Number Theory

• 0 Divided by 0 [Rimshick, 11/27/1997]
  What is the answer to 0 divided by 0? I think it is undefined...

• 100 Factorial in Base 6: How Many Zeros? [Krista, 10/24/2001]
  How many zeros are there at the end of 100! in base 6?

• Adding Arithmetic Sequences [Tanner, 07/10/1998]
  How do you add the numbers from 1 to 5000 without actually doing it or using a calculator? What if you were adding just the odd numbers?

• Adding Hexadecimals [Donaldson, 07/13/1998]
  A complete introduction to the hexadecimal, including sample problems on addition.

• Adding in Base 9 and Base 5 [Costa, 10/21/1997]
  In 3rd grade my son is learning how to add in bases other than 10.

• Arithmetic in Other Bases [Linda, 02/25/2002]
  I need to know how to write out the steps for addition, subtraction, multiplication, and division in other bases.

• Base 2 Subtraction and Clock Arithmetic [Bell, 01/26/2001]
  We know that computers use the "2's complement" method to do subtraction of base 2 numbers, and that this method works every time, but we have no idea why it works.

• Base 12 Fractions [Karla, 11/05/2001]
  I need a simple way to understand how to do and interpret decimals in base 12.

• Base 26 [Foley, 01/27/1997]
  In a base 26 number system where the letters of the alphabet are the digits, calculate TWO+TWO in this system and express the answer in base ten. Express 1997 in base 26.

• Base Number [Steve, 10/07/1998]
  What does base number mean?

• Base Number Equivalence Tables [Laurianne, 10/30/2001]
  My son needs to show tables for base 3 and base 7.

• Binary Conversion [Stirling, 01/07/1998]
  In an adult computer course we are supposed convert binary numbers to real numbers and binary to hexadecimal.

• Binary Subtraction [Houston, 7/25/1996]
  I understand the rules of adding in binary, but how in the world do you subtract?

• Binary System [Rae, 5/22/1996]
  How do you use the binary system?

• Calculating the Last Digit of an ISBN Number [Smith, 07/15/1998]
  How do you calculate the last number in an ISBN number?

• Calculator Addition [Anthony, 6/30/1996]
  How does a calculator add?

• Cardinal and Ordinal Numbers [Aitchison, 01/08/1997]
  How can 3 be both a cardinal and ordinal number at the same time?

• Classifying Numbers [Kelly, 09/06/2001]
  Can you tell me about complex numbers, real and imaginary numbers, rational/irrational numbers? I don't understand any of it.

• Concepts of Adding in Base 2 [Ross, 08/18/1998]
  I don't understand the whole concept of base 2.

• Congruum Problem [Allan, 04/04/2002]
  I have found a reference to Fibonacci and his congruum problem. But something has me stumped...

• Converting from Base 6 to 10 and Back [Leah, 09/26/2001]
  I need to know how to convert 2411 in base 6 to base 10.

• Counting: Base 6, 12, 16 [Asplund, 5/31/1996]
  How do you count in base 6 and base 12? What about base 16?

• Decimal To Fraction Conversion [Lawson, 06/25/1998]
  I am trying to find a method (one that can be programmed on a PC) to convert the decimal part of a real number to a fraction reprsented by integers for the numerator and denominator.

• Defining Kinds of Numbers [Bunting, 03/21/1997]
  Could you please define: perfect numbers, deficient numbers, square numbers, abundant numbers, amicable numbers, and triangular numbers?

• Definition of the Signum Function [Little, 05/31/2000]
  Can you give me a simple definition of the signum function, and any practical examples of its usage?

• Difference Between Zero and Nothing [Cockburn, 12/12/1996]
  What is the difference between zero and nothing?

• Divide by 0 Undefined? [Kloofhs, 9/10/1996]
  When something is divided by 0, why is the answer undefined?

• Dividing by 9, 99, 999.... [Snyder, 9/7/1996]
  Please explain why dividing x by n 9's (x/99, for example) yields a sequence of n repeating digits.

• Divisor Counting [Yu, 9/24/1995]
  A student asks a question about the Math Forum Problem of the Week.

• Erdos' Proof of Bertrand's Postulate [Biermann, 3/23/1997]
  I am looking for a proof of Bertrand's Postulate (there exists a prime number between n and 2n (n>2)), a.k.a. Chebychev's Theorem.

• Even + Even = Even [Courtney, 11/06/2001]
  Prove that an even number + an even number = an even number.

• x Factorial and the Gamma Function [Basham, 05/29/1998]
  What is x! when x is 0, negative, or not a whole number?

• Fermat's Last Theorem - Disproof? [Kumar, 05/09/1998]
  I would like to ask you whether there is any problem with the following disproof of Fermat's last theorem...

• Fermat's Last Theorem: Explanation [Renjen, 11/02/1997]
  Please give me some information on Fermat's Last Theorem.

• Finding and Factoring Large or Mersenne Primes [Cypra, 02/22/1998]
  How do you find extremely large primes (Mersenne Primes) and how do you tell if they are prime? What is the most efficient way of factoring primes?

• First Ten Perfect Numbers [Tink, 12/11/2001]
  I have to figure out at least 10 perfect numbers for my homework.

• Fraction/Decimal Conversion to Other Bases [Trinidad, 08/12/1998]
  What are the rules for converting fractions to binary and octal and vice versa?

• Frequency of Primes [Knobler, 03/17/1997]
  Is there a pattern to describe the frequency of prime numbers?

• How can .999999.... equal 1? [Emily, 03/21/2001]
  If .99999999.... goes on forever, wouldn't it be just a little below one?

• Imaginary Numbers - History and Commentary [Engel, 09/04/1997]
  Some history and a different approach to imaginary numbers; Hamilton's approach.

• The Infamous .999... = 1 [Dan, 01/12/2002]
  Can we give infinity a value?

• Infinite Sets [Lee, 07/17/1997]
  How do you prove that there are more rational numbers than negative integers? How can you tell if an infinite set is countable or uncountable?

• Infinity in Undefined (Divide by 0) Situations [White, 9/5/1996]
  Elasticity is an inverse slope. If a graph of elasticity is a horizontal line, it is referred to as infinitely elastic. But, wouldn't this really be undefined because it is a divide by zero situation?

• Introduction to Bases in Math [Billy, 02/27/2002]
  Rewrite the base 10 numeral in base 5: 13. I don't understand.

• Is 120 a Perfect Number? [Sweigard, 10/15/1997]
  Is 120 a perfect number? It seems to fit all of the criteria.

• A Million Seconds [Alyson, 09/10/2001]
  What is a million seconds in weeks, days, hours, minutes, and seconds? What does unit conversion have to do with number bases?

• Natural Numbers, Positive Integers [Manalastas, 04/07/1997]
  Why is the set of natural (counting) numbers, i.e. {1,2,3,...}, different from the set of positive integers {1,2,3,...}?

• Negative Bases [Wile, 06/10/1999]
  Are there such things as negative bases?

• Numbers [Greeno, 12/10/1997]
  How do integers and whole numbers, rational numbers, transcendental numbers, and counting numbers relate to each other?

• Palindromic Squares [Jimmy, 07/28/1997]
  Do you know any numbers besides 14641 where both the number and its square root read the same left to right as right to left?

• Perfect Numbers [Tahquitz, 10/17/1996]
  Is there any way, other than by trial and error, to figure out what the perfect numbers are?

• Perfect Numbers [Ken, 12/07/1996]
  What are the first 10 perfect numbers? Is there a formula for getting a perfect number?

• Planes and Lines [Tao, 10/26/1996]
  Do planes and lines contain the same number of points?

• Prime Number 2001, Sieve of Eratosthenes [Davenports, 01/25/1997]
  Is 2001 a prime number?

• Primes and Perfect Numbers [Schrodinger, 08/08/1997]
  Are there infinite numbers of prime and perfect numbers?

• Proof by Induction - Pascal's Triangle [Sylvestr, 4/3/1996]
  I was given a proof by my math teacher by the means of mathametical induction: Prove i(nCi) = n2^n-1

• Properties and Postulates [Nwasokwa, 08/04/1999]
  How do you discover or create a property? What is the difference between a property and a postulate? Do we have to prove all properties?

• Prove That -(-a) = a [Eduardo, 09/11/2001]
  How do you prove that -(-a) = a using the properties of real numbers? What about -1 * -1 = 1?

• Pythagorean Triples [John, 11/18/1997]
  Could you explain how pythagorean triples work, are calculated, etc.?

• Real Number Terminology [Silver, 12/04/1996]
  What does it mean to be non-existent over the reals?

• Remainder Problem [Virginia, 10/27/2001]
  What number less than 500 produces remainder 4 when divided by 5, remainder 7 when divided by 9, and remainder 9 when divided by 11?

• Remainders of 1, 2, 3, 4 [Laura, 10/09/2001]
  Find the smallest whole number that when divided by 5, 7, 9, and 11 gives remainders of 1, 2, 3, and 4 respectively.

• Representing Numbers in Different Bases [Irene, 08/05/1998]
  How did they get the 2 x 3 cubed? Plus some number less than 3 cubed?...

• Reversing a Number by Multiplying by 9 [Chan, 08/23/99]
  When some numbers are multiplied by 9, why is the result the reverse of the original number?

• Significant Non-Zero Digits [Sarah, 11/27/2001]
  How many significant digits are there in a number with no non-zero digits? Example: 00.000 Are there any?

• Size of Infinity [Cramer, 07/22/1997]
  How can one infinity be bigger than another infinity?

• Squaring the Circle [Deloach, 12/22/1997]
  Can you construct a square at all with the same area as a circle with a given radius?

• Smallest Value of N!; Factorial Table [Melissa, 11/07/2001]
  If N! ends in exactly 3 zeros, what is the smallest possible value of N?

• Tree Diagram for Math Numbers [Hallock, 10/05/1997]
  My daughter is doing a tree diagram using terms related to math "numbers." Could you please explain in lay terms what surds are?

• Why are Operations of Zero so Strange? [Knobler, 03/17/1997]
  Why do we say 1/0 is undefined? Can't you call 1/0 infinity and -1/0 negative infinity? Why not? What is 0 * (1/0)? What is the quantity 0^0?

• Why Isn't Infinity a Number? [Shefl, 2/15/1995]
  The concept of infinity is not considered a number. Why?

• Why a Zero Exponent Equals One, and Changing Number Bases [Lyons, 9/26/1995]
  Why is any number to the zero power equal to one? Could I have some information on hexadecimal and binary for my classes?

• Why is Zero the Limit? [Katie, 02/25/2002]
  Why is zero called the limit of the terms in the sequence the limit of 1 over n, as n approaches infinity, equals zero?

• One Plus One isn't Two [Huang, 1/10/1995]
  I was once shown that 1+1 isn't 2, and I don't remember how it was done. Could you please e-mail me with an answer?

• Euclid's Proof on the Infinitude of Primes [Goldstein, 10/31/1995]
  Which Greek mathematician proved that there is no greatest prime number?

• Why Does 0! = 1 ? [Meesterr, 12/8/1995]
  Why does 0! = 1 ?

• Binary Numbers [Zaktzer, 12/15/1995]
  A student asks for information about using base two.

• Modulus Operator Problem [Mayes, 2/12/1996]
  Can you explain this problem to me? 4 mod 3 = ?

• Consecutive Non-Prime Integers [Unknown, 2/17/1996]
  For any integer n > 1 there exist n consecutive non prime integers. I have taken n = 4 and then used 24,25,26,27 as a specific example. However, for n large I don't see how this can hold.

• Twin Prime Numbers [Yu, 3/11/1996]
  You know that a prime number is a whole number greater than 1 whose only whole number divisors are 1 and itself. You may not know that there are also such things as twin prime numbers....

• Irrational and Prime Numbers [Mahomes, 4/1/1996]
  What are irrational numbers; is there a highest prime number?

• Number Theory - Perfect Square [Bianchi, 5/26/1996]
  Find all the possible values of n...

• A Theorem to Find Lattice Points [Jryu, 6/1/1996]
  What are the conditions under which the line ax+by=c will contain lattice points?

• Euler's theorem [Pooker, 7/2/1996]
  How do I find the inverse of a modulo m using Euler's theorem?

• Coins in a Square Array [Geldermann, 7/8/1996]
  I left some coins on the table in a square array and now there are only two left... prove that the butler lied.

• Complement of a Number [Steve, 7/10/1996]
  What is the method for finding the complement of a number?

• Zero as an Exponent [Tbui, 7/15/1996]
  Why does n^0 = 1?

• Perfect Number [Tadeu, 7/21/1996]
  How can I find a perfect number? What are some reference books about number theory?

• Prime Numbers between 1 and 150 [Mishra, 7/30/1996]
  How many prime numbers are there between 1 and 150?"

• 1+1 Theorem [Chen, 8/5/1996]
  What is the 1+1 theorem?

• Converting Bases [Blanton, 9/5/1996]
  How do I convert from one base to another - i.e. decimal to bin, hex, and oct?

• Definitions: Relatively Prime, Proper Factor [Fitzpatrick, 9/11/1996]
  What does it mean to be relatively prime? What is a proper factor?

• Dedekind Cuts [Stern, 10/23/1996]
  What is a Dedekind cut?

• Converting from One Base to Another [Geise, 10/28/1996]
  How do you convert a number base 9 to base 4? from base 9 to base 10?

• Formula for Factors of a Number [Daniel, 11/3/1996]
  How many triangles can you draw on a square grid of dots of size x*x?

• Numbers with 12 Factors.... [Conrad, 11/15/1996]
  I have to find two numbers that have exactly 12 factors...

• Programs to Find Prime Numbers [McNeil, 11/27/1996]
  Can a program be written in BASIC to compute the number of prime numbers smaller than n?

• Using Mod to Find Digits in Large Numbers [Paulson, 12/10/1996]
  Find the last two digits in 1996^1996.

• Are All Perfect Numbers Even? [Hazen, 01/16/1997]
  Has it been proved that perfect numbers must be even?

• Indirect Proofs [Quinn, 01/30/1997]
  Give a proof that if r is any nonzero rational number, and s is any irrational number, then r/s is irrational.

• Sizes of Infinities [Clackum, 01/31/1997]
  How can you prove that one infinity is larger than another?

• Different Infinities [Hicks, 02/19/1997]
  How many different infinities are there?

• Integers and Complex Numbers [Brian, 02/27/1997]
  Do hyper-reals and octonions exist outside complex numbers?

• Multiple Personality Numbers [User, 03/13/1997]
  A rectangular array of a number N is the number of rectangular arrays that can be formed from N dots. Of all the numbers less than 1 million, which has the most rectangular arrays and why?

• Proof That 2 Does Equal 1! [Mucha, 03/24/1997]
  I came up with a proof that 1 = 2. Where does my math go wrong?

• Uses of Imaginary Numbers [Boyer, 03/24/1997]
  Can you tell me a real-life application of imaginary numbers?

• Greatest Common Factor [Andy, 03/28/1997]
  How do you find the greatest common factor?

• Binary Operations [Matt, 04/07/1997]
  How do you do binary addition, subtraction, multiplication, and division?

• Zero and Infinity [Sanson, 04/24/1997]
  Why is the quotient of a number divided by zero infinity?

• Roots of ax^f2 bx+c = 0 [Bullock, 05/22/1997]
  Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0 are not rational.

• The Square Root of i [Leif, 05/25/1997]
  What is the square root of i?

• Base 16 [White, 07/07/1997]
  How do you add and subtract in base 16?

• Fundamental Theorem of Arithmetic [Bill, 07/08/1997]
  What's so fundamental about the fundamental theorem of arithmetic?

• Finite vs. Infinite [Angie, 07/10/1997]
  If a line segment is a measurable part of a line, why is the number of points that make up a line segment infinite?

• Pythagorean Triples [Toscano, 07/14/1997]
  Is there a formula to determine the solutions to the following equations? a^2 + b^2 = c^2, a^3 + b^3 + c^3 = d^3...

• Variable Within and Outside an Exponent [Shannonhouse, 07/29/1997]
  Solve for t: d = a*t + b*e^-(c*t) where a, b and c are constants and e is the exponential.

• Real Numbers [Seldon, 08/08/1997]
  What exactly is a real number?

• Probability of Random Numbers Being Coprime [Knobler, 08/12/1997]
  I have heard that the probability of two randomly selected integers being coprime is 6/(pi^2). How do you show this is true?

• Proof that Sqrt(3) is Irrational [Gardner, 08/14/1997]
  How does one prove that sqrt(3) is irrational? or others? Is there a general algorithm? How about just for primes?

• Pairs of Integers [Sandip, 08/16/1997]
  Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.

• Dividing 29/49 [Fredericks, 08/30/1997]
  Can I divide 29/49 out until it repeats itself or terminates without using long division?

• Relatively Prime Pythagorean Triples [Love, 09/13/1997]
  Questions about Pythagorean triples.

• Why is 0! 1? [Kimberly, 09/14/1997]
  Why is zero factorial 1?

• Patterns in Rolling 3 Dice [Schaper, 09/16/1997]
  I have come out with 216 outcomes when rolling 3 dice; while I was listing all the outcomes, I began to see a pattern...

• Indeterminate Forms [White, 09/18/1997]
  What is infinity divided by infinity?

• Casting Out Nines and Elevens [Racela, 09/19/1997]
  Why is nine used in proving this math answer?

• Divisibility of Zero Theory [ONeal, 10/06/1997]
  A student claims that he has heard of divisibility OF zero theory... can you fill me in on this concept?

• Continued Fractions [Gold, 10/07/1997]
  Exactly what are "continued fractions"?

• Help with Proofs [Gelfand, 10/16/1997]
  Could you please help me with some proofs from a course called "Mathematical Thinking"?

• Uncountable Numbers [Cansever, 10/20/1997]
  Why are the real numbers between 0 and 1 uncountable?

• Find the Smallest Number... [Starzyk, 10/21/1997]
  ... that has factors of 1, 2, 3, 4, 5, 6, 7, and 8.

• Zero as Denominator [Amburgey, 10/22/1997]
  Why can't zero be in the denominator for rational numbers?

• Formula for Pythagorean Triples [Sunde, 10/23/1997]
  Is this formula: a = (m^2-n^2); b = 2mn; c = (m^2+n^2) correct for all Pythagorean triples?

• Natural Logarithms [Waissblut, 11/01/1997]
  What's "natural" about natural logarithms? Why is 'e' a transcendental number?

• Fibonacci Formula Inductive Proof [Joly, 11/05/1997]
  I must prove by induction that F(n) = (PHI^n - (1 - PHI)^n) / sqrt5...

• 1000 Lockers [Thorsheim, 11/06/1997]
  The 1st student opens all 1000 lockers, the 2nd student closes lockers 2,4,6,8,10, etc., the 3rd student opens lockers closed and closes lockers open on lockers 3,6,9,12,15...

• Transfinite Numbers [McSwain, 11/07/1997]
  I know that Georg Cantor discovered transfinite numbers, but what are they?

• Divisibility by 37 [Renselaar, 11/08/1997]
  Take a 3-digit number and add to that its "rotation". Prove that the sum can always be divided by 37.

• Non-terminating Decimal Representations of Fractions [Swierzbinski, 11/10/1997]
  Why when you take a finite, limited quantity like one-third and turn it into a decimal do you get .333... on into infinity?

• Graphs - Proving the Infinite Ramsey Theory [Madarasz, 11/10/1997]
  In a graph with infinite "points," if we colour the lines with two colors we'll have either a red or a blue infinite chain of lines, an infinite number of points, all of them joined to each other with the same colour...

• Perfect Numbers [Moreno, 11/10/1997]
  Do the sums of the digits of perfect numbers always equal 1?

• Finding Prime Numbers [Cajuste, 11/10/1997]
  What is the fastest way to determine if a number is prime?

• Synthetic Division [Stuckey, 11/13/1997]
  Why does synthetic division work?

• Finding Pi [Trinh, 11/14/1997]
  What is the quickest algorithm for finding pi?

• Diophantine Equations [Felgate, 11/17/1997]
  We have searched the web for information about diaphantine equations.

• Pythagorean Triples [Matt, 11/19/1997]
  I need to know the first five Pythagorean triples after 3,4,5...

• Proving e is Irrational [Decker, 11/19/1997]
  My professor suggested using a proof by contradiction, but I don't understand how to do it.

• Converting Numbers: Binary to Decimal [Davidson, 11/20/1997]
  I just cannot convert from binary to decimal and back again.

• Logarithms and Base E [Steven, 11/20/1997]
  Why is the base of a natural logarithm "e" - how did "e" receive a value of 2.17... ?

• Finding Formulas for Number Sequences [Chu, 11/22/1997]
  My question is about trying to find a formula between numbers.

• Linear Diophantine Equations [Richers, 11/27/1997]
  ... how was the t-variable introduced, and what is the general method?

• Analytical Solution [Atia, 12/01/1997]
  Can you give me an analytical solution of S[N] = Sum[k^2, {k,1,N}] ?

• De Moivre's Theorem [Pountney, 12/03/1997]
  Prove De Moivres theorem: - (cos(x)+isin(x))^n = cos(nx) + isin(nx) .

• Numbers and Digit Sums [Moltay, 12/03/1997]
  How many numbers between 0 and 99,999 are there whose digits add up to 20?

• Sum of Two Squares [Writz, 12/04/1997]
  What is the smallest number that can be expressed in twelve different ways as the sum of two squares?

• If N is Odd [Waldo, 12/05/1997]
  Prove that if n is odd, then 8 divides (n^2-1).

• Pythagorean Triple with 71 [Lok, 12/07/1997]
  Is there a Pythagorean triple that contains the number 71?

• Even-Digit Palindromes Divisible by 11 [Bob, 12/08/1997]
  Can it be proved that every even-digit palindromic number is divisible by 11?

• Large Prime Numbers [Ware, 12/17/1997]
  Is there an algorithm to determine whether a very large number is prime?

• First Calculation of E [Vlassis, 12/18/1997]
  How did Euler first calculate the value of e?

• Converting Bases [Reinhardt, 12/21/1997]
  How do you convert hexadecimal, binary, and decimal numbers?

• Are There Infinitely Many Perfect Numbers? [Jones, 12/22/1997]
  There are so many theories, but nobody seems to have come up with anything definite...

• Twin Primes [Lee, 12/24/1997]
  Are there any studies being conducted on twin primes?

• Finding the Greatest Common Factor of Two Different Numbers [Knight, 01/06/1998]
  Finding the GCF of two different numbers after using a factoring tree to find their factors.

• Fibonacci Sequences [Arbit, 01/08/1998]
  Please help me with a proof.

• Irrationality of Pi [Chitwood, 01/09/1998]
  Is C/d = a rational number if actually measured?

• Fermat's Theorem [Li, 01/21/1998]
  Why was Fermat's Theorem such a mystery?

• Divisibility of Squares of Prime Numbers [Stryk, 02/14/1998]
  If p is a prime greater than 3, prove that p^2 leaves a remainder of 1 when divided by 12.

• Hexadecimal System [Rhodus, 02/15/1998]
  I know the binary system using base two, but I don't understand the hexadecimal system using base 16.

• The Limit of (1+1/x)^x As x Approaches Infinity [Harvey, 02/17/1998]
  How Euler calculated e, and what it has to do with the equation (1+1/x)^x.

• Unsolvable and Unsolved Problems [Jitsu, 02/19/1998]
  What's the difference between problems like Squaring the Circle and Goldbach's Conjecture or the Collatz Problem?

• Casting Out Nines [Monaco, 02/19/1998]
  I am trying to find a reference which defines this mathematical operation...

• Angstrom Numbers [Basak, 02/20/1998]
  Numbers (0 and 1 excluded) in which the sum of the cube of the digits is equal to the number itself. Is it true that between the numbers 2 and 10000, there are only 4 such numbers?

• The Golden Ratio [Miller, 02/23/1998]
  I know that the limit of the ratios of the Fibonnaci sequence is the golden mean, but I would like to see a proof.

• Primitive Pythagorean Triples [Tsai, 02/23/1998]
  Given a triple of numbers (a, b, c) so that a, b, and c have no common factors and satisfy a^2+b^2 = c^2, make a guess about when a, b, or c is a multiple of 5.

• Trick for Numbers Divisible by 3 or 9 [Planken, 02/24/1998]
  Proof of a trick for numbers that are divisible by 3 or 9.

• Imaginary (Complex) Numbers [Kyle, 02/26/1998]
  What are imaginary numbers?

• Zero Laws and L'Hopital's Rule [Sheryl, 03/04/1998]
  Is zero divided by zero: a) zero, b) undefined, or c) one?

• Divisibility Proof [Itay, 03/09/1998]
  Divisibility of any given positive integer by another built from only 1's and 0's.

• Formula for the First Day of a Year [Leonard, 03/18/1998]
  Is there an equation to find the first day of a year given the year?

• Converting to Base 16; Place Value Chart [Terri, 03/22/1998]
  How do you convert numbers to base 16 numbers?

• Irrationality of Root 2 [Lord, 03/26/1998]
  I've heard that there is a way to prove that root 2 is irrational using a DIRECT method of proof...

• Narcissistic Numbers, Weird Numbers, and Fortunate Primes [Tang, 03/27/1998]
  Definitions and examples of narcissistic numbers, weird numbers, and fortunate primes.

• Perfect Number Algorithms [Nilsson, 03/31/1998]
  What is the formula for a computer program that tests whether an integer is a perfect integer?

• 22/7 as an Approximation for Pi [Faulkner, 04/01/1998]
  Approximating pi by simple continued fractions.

• The Number of Divisors of an Integer [Lloyd, 04/02/1998]
  Formula and proof for the total number of divisors of any integer.

• Trailing Zeros and Zero Factorial [Lopez, 04/07/1998]
  How many trailing zeros are there for 100! ?

• Getting 0.99999... [Dusty, 04/15/1998]
  Is there any mathematical way to get 0.99999999999......?

• Primality Testing [Png, 04/22/1998]
  Is there any fomula to find if a number is a prime?

• Proving Phi(m) Is Even [Damaso, 04/22/1998]
  Explain why phi(m) is always even for m greater than 2...

• Division of Large Numbers [Salazar, 04/28/1998]
  What is the remainder when 7^100 is divided by 13? Give a general strategy and an explanation.

• Finding Patterns in Digits [Spriggs, 04/29/1998]
  How can we find other solutions to problems like 2^5*9^2 = 2592?

• Solving Cubic and Quartic Polynomials [vonBerwick, 04/30/1998]
  Could you describe the algorithms used to solve cubic and quartic polynomials (Tartaglia's Solution)?

• Solving Multivariable Diophantine Equations [Jin, 05/03/1998]
  Finding general solutions to two diophantine equations.

• Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? [Hanna, 05/08/1998]
  Using limits to prove that 1^infinity, infinity^0, and 0^0 are indeterminate forms.

• Lucky Number Sequences [Dellit, 05/11/1998]
  A lucky number is one for which the sum of its digits is divisible by 7. Can you help me find the following patterns?

• Sums of Three Squares [Darcy, 05/18/1998]
  What numbers cannot be expressed as the sum of three squares?

• Lucky and "Elucky" Numbers in Consecutive Numbers [Amy, 05/21/1998]
  Showing that 13 consecutive numbers always contain at least one number the sum of the digits of which is divisible by 7.

• Number and Its Square Using All 9 Digits Exactly Once [Faye, 05/22/1998]
  Using multiplication facts to find all the whole numbers for which the number and its square together use exactly nine digits 1, 2, 3, ..., 9 only once.

• Counting Odd Coefficients [Ehsan, 05/27/1998]
  If (1+x)^100 is multiplied out, how many of the coefficients are odd? How would you generalize?

• What is the Gamma Function? What is Gamma of 4? [Masdeo, 05/28/1998]
  Deriving G(4) = 3! from the gamma function integral.

• Minimizing the Sums of Squares [Hrycenko, 06/12/1998]
  Find two numbers such that their sum is 20, and the sum of their squares is as small as possible.

• Prove that Log A is Irrational [Magictek, 06/14/1998]
  Can you help me prove that a common log of a number (not powers of 10) is irrational?

• Babylonian Number System [Carter, 06/15/1998]
  A sexagesimal (base 60 instead of 10) number system.

• Are Prime Numbers Infinite? [Vincent, 06/18/1998]
  How do we know the number of primes is infinite?

• Happy Numbers [Jarman, 06/21/1998]
  What are happy numbers?

• Chinese Remainder Theorem [Burns, 06/27/1998]
  The teacher has some apples to distribute to her students...

• Intersection of Lines [Andrew, 06/28/1998]
  n co-planar lines are such that the number of intersection points is a maximum. How many intersection points are there? ...

• Triangle Proofs [Behl, 06/28/1998]
  The sides of a triangle are a,b,c; prove that (a+b+c)^3 >= 27(b+c- a)(c+a-b)(a+b-c)...

• Smallest Number Puzzle [Behl, 06/30/1998]
  Find the smallest number which when divided by 9,13,17, and 25 leaves remainders 1,0,2, and 3 respectively.

• Quadratic Residues [Petry, 06/30/1998]
  I need a fundamental explanation of the concept of quadratic residues.

• Knights of the Round Table [Lou, 07/01/1998]
  If x knights are sitting at a round table, and every other one is removed, who is the last one left sitting at the table?

• Proving Perfect Squares [Brisebois, 07/05/1998]
  Suppose a, b, and c are positive integers, with no factor in common, where 1/a + 1/b = 1/c. Prove that a+b, a-c, and b-c are all perfect squares.

• Triangular Numbers [Frances, 07/07/1998]
  How do you know a number is triangular? How is n/2(n+1) derived?

• Prime Numbers [Anderson, 07/10/1998]
  If you multiply all the prime numbers up to N together, the limit appears to be exp(N) as N gets large. Is there a simple reason for this?

• Showing Divisibility [Mills, 07/12/1998]
  How do you show that 5^(2n) + 3(2^(2n+1)) is divisible by 7?

• Proving the Square Root of a Prime is Irrational [Smothers, 07/15/1998]
  How do you prove that if p is prime, the square root of p is irrational?

• Summing a Binary Function Sequence [Ping, 07/16/1998]
  How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where B(n) denotes the sum of the binary digits of n?

• Nested Square Roots [Natasha, 07/17/1998]
  Solve for n where n = sqrt(6 + sqrt(6 + sqrt6 + ...

• Which Fractions Repeat? [Melissa, 07/21/1998]
  How do you know whether a fraction will be a repeating or terminating decimal? If repeating, how many decimal places?

• Amicable Partners [Keelan, 07/23/1998]
  Find the amicable partner of 1184. Choose another number between 100 and 1000 and show that it does not have an amicable partner.

• Monkeys Typing Shakespeare: Infinity Theory [Bridge, 08/05/1998]
  Would an infinite number of monkeys typing at random eventually produce the entire works of Shakespeare?

• Congruence of Integers [Irene, 08/10/1998]
  Can you help me find the remainder when 5 to the power of 1001 is divided by 6...?

• Factoring 13 with Complex Numbers [Scomazzon, 08/11/1998]
  How do you show that 13 is not prime using imaginary numbers? We know that 13 = (3 + 2i)(3 - 2i), but how do you do this in general?

• Paint Formulas in Base 48 [Behymer, 08/20/1998]
  I work for a paint store where our formulas are based on an ounce being 48 parts...

• Summing Consecutive Integers [Simon, 08/30/1998]
  Express 1994 as a sum of consecutive positive integers, and show that this is the only way to do it.

• The Square and Multiply Method [Ravi, 08/31/1998]
  Solve an encryption problem by solving the math function 33815^(81599) (mod 154381).

• Digit Patterns of the Powers of 5 [Peterson, 09/14/1998]
  Why is there a pattern in the last digits of the powers of 5?

• Rectangular Solids from Blocks [Purins, 09/25/1998]
  How many rectangular solids can be made from "n" cube-shaped blocks?

• A Circular Massacre [Kurczak, 09/25/1998]
  Ten thousand sailors are arranged in a circle; starting with the first one, every other sailor is pushed overboard ....

• Last Digits and High Exponents [Wernhoff, 09/26/1998]
  Calculate the following expressions without a calculator: 3^1000 and 7^ 134. In each, what is the final digit?

• Fermat's Little Theorem and Prime Numbers [Levitt, 09/28/1998]
  Please explain how to use Fermat's Little Theorem to test whether a number is composite.

• The Number of Zeros in a Factorial [Jeff, 10/01/1998]
  How many zeros come after the last non-zero digit of 20,000,000! ?

• Prime Numbers in Different Bases [Diaz, 10/07/1998]
  Are all prime numbers the same in all bases? If 21 is a prime, are 10101 (in binary), and 15 (in hexadecimal) also primes?

• The Origin of Lucas Numbers [Smith, 10/08/1998]
  I need help with Lucas Numbers - how and why they were created.

• The Square Root of n! [Johnson, 10/14/1998]
  For what natural numbers n is the square root of n! an integer?

• Bell Numbers [Leonardi, 10/15/1998]
  How are Bell numbers generated? What are the first 12 Bell numbers?

• Four-Digit Palindromes [Evans, 10/21/1998]
  Why is every four-digit palindrome divisible by 11?

• Factoring Large Numbers [McGrew, 10/26/1998]
  Can you give me an algorithm for factoring large numbers? What about the Pollard Rho Factoring Algorithm?

• Explaining the Euclidean Algorithm [Megan, 10/27/1998]
  In the Euclidean Algorithm (or the Division Algorithm), why is the last divisor the greatest common factor?

• Quadratic Residues and Sums of Squares [Meng, 10/28/1998]
  In one of the lemmas in number theory, if p is an odd prime number, then there exist x, y such that x^2+y^2+1=kp...

• Prime Number Tests [Natalie, 11/12/1998]
  Is the number 55409243 prime? How can you test to see whether a number is prime?

• Amicable Numbers [Sparks, 11/12/1998]
  What are amicable numbers? Can you give me examples? What's their history?

• Definitions of Advanced Concepts [Morrison, 11/13/1998]
  Can you give me definitions for: Pythagorean Triplets, Principle of Duality, Euclid's Elements, Cycloid, Fermat's Last Theorem?

• Partitioning an Integer [Huckin, 11/14/1998]
  How many different ways are there of making a number by adding different combinations of three numbers?

• A Hundred-Row Number Pyramid [Williams, 11/19/1998]
  Starting with two(1,2) in the first row of a pyramid and adding one more as you go down the list, what is the last number on the righthand side in the 100th row?

• The Phi Function [Browne, 11/21/1998]
  What are the conditions on n,m so that phi(n*m) = phi(n)*phi(m)? What is phi(p^n*q^m)?

• Summing n^k [Kijjaz, 11/24/1998]
  Is there a general formula for summing the n^k, where k is a positive integer?

• Forming Palindromic Numbers [Keckes, 12/04/1998]
  Can you give me some examples of forming palindromic numbers with different operations? How many steps would it take?

• Computing a^((N-1)/2) mod N [Perriello, 12/07/1998]
  Is there a shortcut for doing a^((n-1)/2) mod N?

• Binary to Hexadecimal [Hamilton, 12/08/1998]
  Is there a simple way to convert from binary (base 2) numbers to hexadecimal (base 16) numbers?

• Primes and Repeating Unit Numbers [Nichol, 12/09/1998]
  How do you prove this statement: For every prime number there exists a repeated unit number that is a multiple of that prime.

• The Zero Power of Two [Burns, 12/10/1998]
  Why is 2 to the 0 power equal to 1? I don't understand how a number can be multiplied by itself zero times.

• Fermat's Last Theorem for n = 3 [Guillon, 12/14/1998]
  What is the proof for Fermat's Last Theorem where n = 3? Who is given credit for the first proof for this case?

• Triangle Perimeters [Olmstead, 12/15/1998]
  How many triangles with integer sides have a given perimeter? How does the triangle inequality enter into the proof?

• Sigma Notation [Pyatakov, 12/17/1998]
  Some summation formulas; finding Sum((n+1)^2).

• Pythagorean Quadruplets [Basias, 12/28/1998]
  I am trying to find a formula that generates Pythagorean quadruplets a,b,c,d such that a^2 + b^2 + c^2 = d^2.

• Prime Number Theorems [Levine, 01/03/1999]
  Can you explain the prime number theorem, Mersenne primes, the Lucas- Lehmer test, and the Riemann Hypothesis?

• Non-integer Powers and Exponents [Tarring, 01/06/1999]
  How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?

• Conjectures vs. Hypotheses [Lau, 01/12/1999]
  What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture?

• Adding and Multiplying to Get 7.11 [Sam, 01/13/1999]
  What four monetary values, when added or multiplied, equal $7.11? (Find A, B, C, D, such that A+B+C+D = A*B*C*D = 7.11.)

• Properties of the Phi Function [Joyce, 01/19/1999]
  What are some properties of the phi function? What about the phi function and prime numbers?

• Fermat's Factorization Method [Karthick, 01/29/1999]
  Can you describe Fermat's method of factoring an integer?

• Percentages of Prime Numbers [Denise, 01/31/1999]
  Does the percentage of prime numbers at every power of 10 decrease until it reaches a constant value?

• Pattern in Period [Michael, 01/31/1999]
  Is there any pattern in odd periods with 1 as numerator?

• Odd Digits of Square Numbers [Bryceland, 02/07/1999]
  Why are there no square numbers other than 1 and 9 that consist entirely of odd digits?

• Factoring [Kevin, 02/09/1999]
  Find the smallest number (integer) that has 30 factors.

• Consecutive Integers [Lori, 02/10/1999]
  Can the product of four consecutive integers be a perfect square?

• Four-digit Palindromes Divisible by 11 [Bonnie, 02/10/1999]
  Why are four-digit palindromes divisible by 11?

• Connecting the Dots [Tim, 03/14/1999]
  If you have a few dots on a page, how many lines does it take to connect them all to each other?

• Subsets and Greatest Common Divisor [Megan, 03/26/1999]
  A question on subsets and another on greatest common divisor (GCD).

• Abundant Numbers [Ashley, 03/27/1999]
  I need to find all the perfect, abundant, and deficient numbers from 1 to 50.

• Simple Example of Ramanujan's Work [Paul, 03/28/1999]
  Ramanujan's contributions to the divisibility properties of partitions of whole numbers.

• Public Key Encryption [Jonathan, 03/29/1999]
  Examples and discussion of operations used for encryption, including mod.

• Binary Divisibility by 10 [Anna, 04/07/1999]
  How can you tell if a binary number is divisible by 10?

• Babylonian Reciprocals (Base 60) [Bobek, 04/08/1999]
  What is the reciprocal of the fraction 451/15 ?

• "Accidental Order" in Pi, e [Ngeno, 04/12/1999]
  Using a result of Dirichlet's to prove that a given sequence will appear in infinitely many prime numbers.

• Repeating Decimals [Grabowski, 04/28/1999]
  I am interested in finding longer repeating groups in number tails of repeating decimals.

• Repeating Digits of Fractions [Grabowski, 04/28/1999]
  Do you know any theorems relating to the length of the repeating portion of the decimal representation of fractions?

• Gaussian Integers [Sandstrom, 05/12/1999]
  Are all real prime numbers also Gaussian prime numbers?

• Mod [Grabowski, 05/17/1999]
  What does the term "mod" mean?

• Pythagorean Triples [Smith, 05/22/1999]
  What is the general formula for all sides of any triple?

• Paths to Triangle Points [Goldstein, 05/26/1999]
  How can I find the number of paths to a point using Pascal's triangle?

• Stirling Numbers [John, 05/26/1999]
  Can you show how to evaluate Stirling Numbers of the first and second kinds?

• Pythagorean Triples [Lea, 05/31/1999]
  Is there a procedure for finding Pythagorean triples?

• Ramsey's Theorem and Infinite Sequence [Chan, 06/01/1999]
  Ramsey's Theorem applied to divisibility in infinite sequences.

• A Pyramid of Layered Marbles [Chris, 06/02/1999]
  How can I find the number of layers, the number of marbles, and the size of a container containing a pyramid of layered marbles?

• Proof that an Even Number Squared is Even [Jason, 06/02/1999]
  How do you prove that any even number squared is even and any odd number squared is odd?

• Coprimes in Fermat's Last Theorem [Oliver, 06/03/1999]
  Why are (z-x)/2 and (z+x)/2 coprime in Fermat's Conjecture when n = 2?

• Duotrigesimal (Base 32) Numbers [deBoer, 06/11/1999]
  A unique and interesting use for base 32 or "duotrigesimal" numbers.

• Converting Binary to and From Decimal [Shroff, 06/16/1999]
  Can you show me how to convert base 2 numbers into base 10 and vice-versa?

• Primes that are Sums of Primes [Tikotekar, 06/22/1999]
  Is there an nth prime number, p, (other than 5, 17 and 41) that is equal to the sum of the prime numbers up to n? For example, the 7th prime is 17=2+3+5+7.

• Number Theory Proofs [Ki, 06/24/1999]
  How can I prove that the equations (x,y) = g and xy = b can be solved simultaneously if and only if g^2|b for integers g, b?

• Reversed Digits Theorem [Tikotekar, 06/24/1999]
  For a positive integer abc..., if (abc...)^n = xyz... and if (a+b+c+...)^n = x+y+z+..., how can I prove that (...cba)^n = ...zyx?

• Program to Convert Number Bases [Manson, 07/12/1999]
  Is there an easier method for converting bases than dividing and collecting the remainders? I want to write a computer program to do this.

• Two's Complement [Corbin, 07/13/1999]
  What is two's complement and how is it used?

• Square Root of a Prime [Moran, 07/14/1999]
  Suppose p is a prime number. Show that sqrt(p) is irrational.

• Floating-Point Binary Fractions [Mairaj, 07/19/1999]
  How can you represent fractions such as 12.93 in binary? How do computers represent such numbers?

• Comparing Numbers in Different Bases [Morrison, 07/28/1999]
  If I am given a number in base 4 and asked if it is bigger or smaller than another number in another base, should I always convert both to base 10?

• Running Time of an Insertion Sort [Thomas, 08/06/1999]
  How can I write and solve a recurrence formula for the running time of an insertion sort? Which is better, an insertion sort or a merge-sort?

• Determining Factors of a 3998-digit Number [Goorman, 08/11/1999]
  Let N = 111...1222...2, where there are 1999 digits of 1 followed by 1999 digits of 2. How can I express N as the product of four integers, each of which is greater than 1?

• Multiplicative Order [Mee, 08/13/1999]
  What is the multiplicative order of 2 mod 2n+1? Can you explain the concept of multiplicative order?

• Prime Numbers in Cryptography [Rebernik, 08/14/1999]
  What are some practical uses of prime numbers?

• Mean of a Set of Numbers by Subsets [Bartlett, 08/15/1999]
  How can I prove that the mean of a finite set of numbers is the mean of the means of all the non-empty subsets of that set?

• Sum of Digits Divisible by 11 [Anoop, 08/16/1999]
  Can you prove that in a sequence of 39 consecutive natural numbers there exists at least one number such that the sum of its digits is divisible by 11?

• Negative Numbers in Binary [Akella, 08/19/1999]
  How do I represent -53 as a binary number? Is it 110101? Wouldn't that be +53?

• Significance of Irrational Numbers [Ukachukwu, 08/23/1999]
  What exactly is the meaning of .333... or pi? What's the difference between point three repeating and point three to the 105th decimal place?

• Casting Out Nines [Fowler, 08/26/1999]
  Can you explain how to use the 'casting out 9's' technique in a way that a 6th grader can understand?

• Triangular Numbers That are Perfect Squares [Manuel, 09/07/99]
  How can I find and prove a general formula which will give me numbers which are both triangular numbers and perfect squares?

• Infinity Hotel Paradox [Evening, 09/15/1999]
  How can a hotel with an infinite number of rooms, all already occupied, accommodate the passengers of an infinite number of buses without doubling them up?

• Primes That Are the Sum of 2 Squares [Jennelle, 09/17/1999]
  How can I prove that every prime of the form 4m + 1 can be expressed as a sum of two squares?

• Prime Factors of 4,194,305 [Offutt, 09/20/1999]
  How can I find the prime factors of 2^22+1?

• Even or Odd in Base 5? [Susan, 09/23/1999]
  Is there a way to find whether a number written in base 5 is even or odd without first converting it to base ten?

• Proof by Mathematical Induction [James, 09/24/1999]
  Prove the following statement by mathematical induction: for any integer n greater than or equal to 1, x^n - y^n is divisible by x-y where x and y are any integers with x not equal to y.

• What is 0.999... + 0.999...? [Jarman, 09/27/1999]
  How can I show that 0.999... + 0.999... = 2?

• Finding a Series Given the Sum [Mike, 09/27/1999]
  How can I find all series of consecutive integers whose sum is a given value x?

• Positives and Negatives with Infinity and Zero [King, 10/05/1999]
  Are there such things in math as +0, -0, and unsigned 0; and +infinity, -infinity, and unsigned infinity? Are these different?

• Newton's Method and Continued Fractions [Kaluhiokalani, 10/06/1999]
  Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

• Relatively Prime [Dingus, 10/07/1999]
  What does the term relatively prime mean, and how can you determine if two numbers are relative primes?

• Operations in Nondecimal Bases [Paley, 10/16/1999]
  How can you subtract, multiply, and divide numbers in other bases?

• Normal Numbers [Rapella, 10/19/1999]
  Is there a non-probabilistic proof of the existence of normal numbers? Is there an algorithm to produce a number r that is normal in two or more different bases? What about irrational numbers like pi, e, or the square roots of 2,3,5, ...?

• Largest x, x^2 less than 2 [Michael, 10/23/1999]
  Prove that there is no largest real number x, such that x^2 is less than 2. (Use indirect proof.)

• Finding Howlers [Nagpal, 10/25/1999]
  Howlers are fractions like 16/64; when you cross out the 6 on the top and the bottom, you are left with 1/4, which is the simplified fraction. How can I find all 2-digit, 3-digit and 4-digit howlers?

• Three Number Theory Questions [Christopher, 10/25/1999]
  Find the sum of the digits in 4444^4444; find how many times the digit 1 occurs from 1 up to 10,000,000,000; find 3 integers greater than 5^100 that are factors of (5^1985)-1.

• Sum of Squares of Two Odd Integers [Devanshi, 10/26/1999]
  How can I prove that the sum of the squares of two odd integers cannot be a perfect square?

• Least Common Multiple [Boyce, 10/26/1999]
  What is the smallest number that is divisible by the numbers 1 through 10?

• C++ Program to Convert Decimal to Binary [Kaufman, 10/26/1999]
  Can you show me a C++ routine that converts decimal numbers to binary?

• Divisibility Proof [Pat, 10/26/1999]
  How can I prove that (n^5-n) is divisible by 30, and (n^7-n) is divisible by 42, without using induction?

• Numbers with the Digit 3 [Namita, 10/27/1999]
  In how many numbers between 1000 and 9999 does the digit 3 occur?

• Average Age at a Party [Amure, 10/27/1999]
  How can I find b+g if the average age of b boys is g, and the average age of g girls is b, and the average age of everyone, including the 42-year-old teacher, is b+g?

• Product of Two Primes [Amure, 10/27/1999]
  How many positive integers less than 100 can be written as the product of the first power of two different primes?

• Divisibility Rule for All Divisors [Chung, 11/07/1999]
  Is there a theorem for figuring out divisibility rules for all natural numbers?

• Primes of the Form 4n+3 [Troise, 11/07/1999]
  Prove that there are infinitely many primes of the form 4n+3 where n is an element of the natural numbers.

• Adding Rational and Irrational Numbers [Troise, 11/07/1999]
  How can you prove that a rational number added to an irrational number results in an irrational number?

• Recurrence Relation for a Pell Equation [Cash, 11/09/1999]
  Can you help me find a recurrence relation for generating solutions to the Pell equation x^2 - 5y^2 = 1?

• What is Octal? [Griffin, 11/30/1999]
  What is octal?

• Why Aren't There Negative Prime Numbers? [Lopez, 12/10/1999]
  Why can't negative numbers be prime numbers?

• Catalan Numbers [Bradley, 12/15/1999]
  What are Catalan numbers and what applications do we have for them?

• Factorials Can't Be Squares [Datta, 02/11/2000]
  Can you prove that the factorial of a number (greater than 1) can never be a perfect square?

• Proving the Properties of Natural Numbers [Harsha, 03/08/2000]
  How can you prove or derive the commutative, associative, and distributive properties of numbers?

• Formula for Counting Triangles [Sebastian, 03/16/2000]
  How many equilateral triangles of integer-length sides are in an equilateral triangle n units on a side?

• Increasing and Decreasing Subsequences; Pigeonhole Principle [Elizabeth, 03/21/2000]
  How can I prove that there exists an increasing OR decreasing subsequence of length n+1 or more in any list of (n^2)+1 distinct integers?

• Generating Function of Catalan Numbers [Chris, 04/04/2000]
  Can you explain the recurrence relation for the Catalan numbers?

• Hexadecimal Subtraction and Multiplication [Schnurr, 04/06/2000]
  How do you subtract or multiply hexadecimal (base 16) numbers?

• Proof That Equation Has No Integer Roots [Liu, 05/09/2000]
  How can I prove that if p is a prime number, then the equation x^5 - px^4 + (p^2-p)x^3 + px^2 - (p^3+p^2)x - p^2 = 0 has no integer roots?

• Even - Odd Handshake Problem [Cox, 05/11/2000]
  How can I prove that the number of persons who have shaken an odd number of hands is even?

• Long Division in Binary [Ames, 05/16/2000]
  How can you divide 1011 base 2 by 11 base 2?

• Using a Calculator in Other Bases [Ames, 05/16/2000]
  Can a calculator be used to add non-decimal numbers? For example, 27 octal + 65 octal.

• Absolute Values and Imaginary Numbers [Khaine, 05/17/2000]
  Could the solution to |x|= -8 be an imaginary number? Since no absolute value can be negative, this [like sqrt(-1)] cannot be solved.

• Proving Fermat's Last Theorem for N = 4 [Dabrowski, 05/18/2000]
  How can you prove Fermat's Last Theorem for the specific case n = 4?

• Fixed Point and Floating Point Numbers [Huband, 05/19/2000]
  What are fixed point or fixed decimal numbers? How do they differ from floating point numbers?

• Proof That the Cube Root of 3 is Irrational [Singh, 05/22/2000]
  How can I show that the cube root of 3 is irrational?

• Prefix for 10^30 Bytes [Boden, 05/25/2000]
  What do you call 1,000,000,000,000,000,000,000,000,000,000 or 10^30 bytes?

• Factoring Large Numbers [Lozano, 05/26/2000]
  How can you use Fermat's Little Theorem to factor large numbers?

• Subtraction Using Nine's and Ten's Complements [Artus, 05/27/2000]
  How does subtraction using the "method of complements" work? Why does it give the correct answer all of the time?

• Floor and Ceiling [Veena, 05/28/2000]
  What do 'floor' and 'ceiling' mean in mathematics?

• Are All Infinitely Long Repeating Numbers Even? [Huggins, 06/06/2000]
  Given an infinitely long repeating series, x = 12341234..., then 10000x = 123412341234... Since 9999 is odd and 12340000... is even, can we say that x is even, and therefore all infinitely long repeating series are even?

• Long Division in Binary [Jindani, 06/06/2000]
  What is the algorithm for binary division? Can you show me some examples?

• Digital Computers and Binary [Marcia, 07/02/2000]
  How do digital computers use the binary number system?

• Search for the Largest Prime [Kochanski, 08/01/2000]
  What is the largest finite number that has a practical use in some branch of mathematics or science? What is the largest prime number known?

• Fermat's Little Theorem [Heden, 09/02/2000]
  Can you help me prove Fermat's Little Theorem, that the expression n^p-n, where p is an arbitrary prime and n is a positive integer, is always divisible by p?

• Paradox in the Zero Power [Zeliger, 09/26/2000]
  Why does the product of multiplying a number by itself zero times equal 1?

• Fundamental Theorems [Rez, 10/02/2000]
  What are the fundamental theorems of algebra and arithmetic?

• Square Roots in Binary [Watkins, 10/03/2000]
  Can you show an example of taking the square root of a binary number?

• Towers of Hanoi [Rakesh, 10/08/2000]
  Can you prove the formula 2^n - 1 for the least number of moves it takes to move all n discs to another peg in Towers of Hanoi?

• Do Rational and Irrational Numbers Alternate? [Cooper, 10/13/2000]
  If any two non-equal real numbers "contain" an irrational, and any two non-equal real numbers "contain" a rational, do rational and irrational numbers alternate?

• Induction With Binomial Coefficients [Lee, 10/16/2000]
  Prove that the sum from i = 1 to n of (i+k-1 choose k) equals (n+k choose k+1).

• Induction on .999... [Thomson, 10/19/2000]
  In the FAQ proof that .999... = 1, how can you multiply .999... by 10 if you can never get to the furthest right value? Can you show me an induction proof that this works?

• Is -1 Prime? [Nesbitt, 10/20/2000]
  -1 has exactly two factors (1 and -1). Does anyone consider it a prime?

• Antifirst Numbers [Karol, 10/23/2000]
  An antifirst number is a number with more divisors than every number before it. I need to write a program that will calculate all the antifirst numbers between 1 and 2,000,000,000.

• Multiplication Using +, -, and 1/x [Silverman, 10/25/2000]
  Suppose you had a calculator with only the +, -, and 1/x operator buttons. Could you do multiplication with it?

• Fermat's Last Theorem with Negative Exponents [Openshaw, 10/26/2000]
  Are there any solutions of Fermat's Last Theorem, x^n + y^n = z^n, for n less than 2?

• Infinity as a Skolem Function [Zaba, 10/28/2000]
  Is infinity an absolute concept, a relative concept, or both?

• Continued Fraction for Tan(x) [Joseph, 11/03/2000]
  Can you tell me why the "infinite continued fraction representation" for the tangent function works?

• Tribonacci Numbers [aznx, 11/11/2000]
  Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers?

• Proof of the Rational Root Theorem [Bucksbaum, 11/13/2000]
  How can I prove the Rational Root theorem?

• The Official Euclidean Algorithm [Julie, 11/16/2000]
  Can you state briefly the "official" Euclidean Algorithm?

• Pythagorean Triples Divisible by 5 [Essex, 11/17/2000]
  Do all right triangles with integer side lengths have a side with a length divisible by 5?

• Sum of 1/Sqrt(i) [Khoury, 11/20/2000]
  What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show me the proof by induction?

• Proof of Lagrange's Theorem [Tse, 11/23/2000]
  I am looking for a proof of Lagrange's Theorem, which states that any positive integer can be expressed as the sum of 4 square numbers.

• Ordering Real Numbers [Ray, 12/11/2000]
  Can you describe how to order real numbers?

• Proving a^x = a^y iff x = y [Lauren, 12/13/2000]
  How can I prove that a^x = a^y iff y = x for all real numbers x and y?

• Prime Numbers [Dick, 12/15/2000]
  How can I prove that if A = (P1*P2+1)^4 - 1, where P1 and P2 are two distinct primes; then A is a multiple of three distinct primes?

• Finding Divisibility Rules for Large Numbers [David, 12/21/2000]
  Is there any system for finding divisibility rules for any number?

• The Indeterminate Nature of 0/0 [Rob, 12/21/2000]
  I have a theory that 0/0 = any number, and is not "indeterminate" as is traditionally claimed. Can you explain the flaw in my thinking, and the "indeterminate" nature of 0/0?

• Sums of Consecutive Integers [Wiltfong, 01/03/2001]
  How many different ways can 2000 be expressed as the sum of two or more consecutive positive integers?

• Sets N, R, C, Z, and Q [Lili, 01/22/2001]
  What are the exact and extensive definitions of the sets N, R, C, Z and Q? What relation do these sets bear to one another?

• Stones, Prime Powers, Induction Proof [Tinka, 01/23/2001]
  A heap of 201 stones is divided in several steps into heaps of three stones each...

• Odd Perfect Numbers [Billingham, 01/23/2001]
  Is this a proof that there are no perfect numbers?

• Diophantine equations in Number Theory [Harp, 01/24/2001]
  If a and b are relatively prime positive integers, prove that the Diophantine equation ax-by = c has infinitely many solutions in the positive integers.

• Proving O(n) [Marriott, 01/23/2001]
  How would you prove that an equation is of order n, or n squared?

• Fundamental Theorem of Algebra [Nataria, 01/25/2001]
  What exactly is the Fundamental Theorem of Algebra?

• How are Binary Codes Used? [Bisset, 01/25/2001]
  I've figured out how the binary system works and how to 'translate' from binary to decimal codes, but how are binary codes used?

• Large-Number Binary Conversion [Godden, 01/25/2001]
  How do you convert very large binary numbers like 2^50 to base 10?

• Fibonacci Sequence [Winters, 01/29/2001]
  Is there a formula for the n-th Fibonacci number?

• Fibonacci Proof [Klein, 01/29/2001]
  This proof is giving me major problems: F(2n) = (F(n))^2 + (F(n-1))^2. ...

• Defining 0/0 [Howard, 01/29/2001]
  I convinced my teacher that 0/0 must be defined, since our math laws say that anything divided by itself equals 1. Shouldn't 0/0 = 1?

• Countability of Rational and Irrational Numbers [Whitehead, 01/30/2001]
  When speaking of countability of numbers, which has more, rational or irrational?

• Fermat Number Proof [Robyn, 01/30/2001]
  Prove that if n is greater than 0, then the Fermat number 2^2^n + 1 is of the form 9k-1 or 9k-4. Prove that n and 2^2^n + 1 are relatively prime for every n greater than 0.

• Generalized Definition of Prime Numbers [Narayanan, 02/01/2001]
  Examining an extended definition of a prime number.

• Induction Proof of Series Sum [Jigar, 02/03/2001]
  How can I prove that for all n greater than 2, the sum 1/(n+1) + 1/(n+2) + .. + 1/(2n) is greater than or equal to 7/12?

• Sums of Consecutive Integers [Kasey, 02/04/2001]
  What numbers can be expressed as the sum of a string of consecutive positive integers?

• Error: Division by Zero [Casey, 02/12/2001]
  How can I explain to my third grader that a number divided by zero is undefined? The school calculator gives the answer 0/E, and the Windows calculator gives positive infinity.

• Divisibility Proof [Promise, 02/16/2001]
  How can I prove that if n is an odd positive integer, then 2269^n + 1779^n + 1730^n - 1776^n is an integer multiple of 2001?

• A Number Digits Puzzle [Promise, 02/23/2001]
  How can I determine all positive integers with the property that they are one more than the sum of the squares of their digits?

• Proving the Associative Property [Valverde, 02/24/2001]
  How can I prove that a binary operation is associative, if all I am given is a table for the operation?

• LaGrange's Theorem [Joan, 02/24/2001]
  Please explain LaGrange's Theorem on the number of roots of a polynomial.

• Finding N Consecutive Composite Numbers [Parks, 02/26/2001]
  How can I find N-1 consecutive numbers that are not prime for any number N greater than 1?

• Real and Rational Numbers [Eileen, 02/27/2001]
  How can I show that the number of rational numbers between 0 and 1 is the same as the number of natural numbers (considering the ordering of fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?

• Lagrange's Theorem [Chapman, 02/27/2001]
  In your archives you show proofs of Lagrange's theorem that every positive integer can be expressed as the sum of four squares, but is there an algorithm for identifying which four squares?

• Sums of Consecutive Positive Integers [Loulie, 03/02/2001]
  Why are the powers of 2 are the only numbers you cannot get as the sum of a series of consecutive positive integers?

• The 'First to 100' Game [Shanthi, 03/12/2001]
  Two players take turns choosing any number from 1-10, keeping a running sum of all the numbers. The first player to make this sum exactly 100 is the winner. Is there a surefire way to win this game?

• 36 Sums, Blank Dice [Kim, 03/15/2001]
  You have two blank, six-sided dice, and you can put any numbers on them. The 12 numbers you choose should allow you to make the sums from 1-36...

• Generating Pythagorean Triples [Samantha, 03/17/2001]
  I need to generate the sixteen primitive Pythagorean triples, and to find how many there are such that the numbers of the triplet lie between 1 and 100.

• Last Four Digits of 5^64 [Walmer, 03/27/2001]
  How can I find the last four digits of 3^125 or 5^64?

• Proof That Product is Irrational [Ellen, 03/28/2001]
  How can I prove that the product of a non-zero rational number and an irrational number is irrational without using specific examples?

• Converting Directly from Base to Base [Ahmed, 04/01/2001]
  Is there a way of reducing the representation of a natural number from one base to another directly without reducing to base 10 first?

• Diagonal Sum in Pascal's Triangle [Potter, 04/02/2001]
  Find the sum of the reciprocals of the diagonals in Pascal's triangle.

• Integer Solutions of ax + by = c [Armentrout, 04/03/2001]
  Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers?

• Frequency of Digits in Pi [Kate, 04/05/2001]
  What digit occurs least frequently in pi?

• Writing Numbers in Bases Greater Than 10 [Cindy, 04/05/2001]
  What would 4 x 13 [base 10] look like in base 42? Do all bases above ten use the same method?

• Base of Roman Numerals [vanCuylenberg, 04/07/2001]
  What is the base of the Roman numeral system, base 10 or base 5?

• Second-Degree Two-Variable Diophantine Equation [Melvin, 04/12/2001]
  Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.

• Infinity Solution [Chen, 04/12/2001]
  Can infinity be the solution to the equation 1 + 2x = 3 + 2x?

• Prove that n^3 + 2n is Divisible by 3 [Jenny, 04/15/2001]
  Prove by mathematical induction that n^3 + 2n is divisible by 3 for all positive integers.

• Arithmetic/Geometric Mean Inequality Theorem [Jeff, 04/15/2001]
  Prove the AM-GM (arithmetic mean - geometric mean) inequality theorem (prove that (x1+x2+x3+...+xn)/n is greater than or equal to (x1*x2*x3* ...*xn)^(1/n).

• Line and Unit Circle; Pythagorean Triples [Kennedy, 04/16/2001]
  If (X,Y) is a point in the 1st quadrant on the unit circle and m is the slope of the line passing through (X,Y) and the point (0,-1), how can I express the coordinates (X,Y) in terms of m? Can this be used to generate Pythagorean triples?

• Modulus Proof [Ooi, 04/16/2001]
  Can you please show me why m^(2^n) = 1 mod(2^(n+2)) when m is an odd integer?

• Remainder when Dividing Large Numbers [Ooi, 04/17/2001]
  How can I find the remainder when (12371^56 + 34)^28 is divided by 111?

• Modulus Congruence Proof [Ooi, 04/18/2001]
  How can I prove 2^(3n+2)+21n = 4 mod (49)?

• Simultaneous Modulus Congruencies [Ooi, 04/18/2001]
  How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?

• Indeterminate Forms [Brian, 04/23/2001]
  Concerning the indeterminate forms such as 0/0 and infinity/infinity, why is one to the infinite power considered an indeterminate form?

• Proof That sin(5) is Irrational [Howard, 04/24/2001]
  How do you prove that sin(5) is an irrational number?

• Coefficients in a Trinomial Expansion [Birkenfeld, 04/24/2001]
  In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

• Irrationality Proof [Brian, 04/26/2001]
  I need to show that log 2 base 10 is irrational.

• Infinity to the Zero Power [Fady, 04/28/2001]
  Does (infinity)^0 equal 1? Why or why not?

• Closed Operations for Negative Irrationals [Lisa, 04/28/2001]
  What set of operations is closed under negative irrational numbers?

• Calculators and Irrational Numbers [King, 05/02/2001]
  When I square the square root of 11 on any calculator, I get the answer 11 (exactly). That seems to indicate that the square root of 11 is a rational number, but it's not. Can you explain this?

• System-Level Programming and Base 2 [Eric, 05/03/2001]
  In computer programming, I have a result that contains several values, always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304, how can I tell if 2^3 exists in 304?

• Primes and Squares [Ali, 05/03/2001]
  For what values of prime number p is (2^(p-1)-1)/p a perfect square?

• Square Root of 2 as a 'Vulgar Fraction' [Dave, 05/04/2001]
  Can the square root of 2 be expressed as a fraction?

• Problem Posed by Fermat [Paul, 05/04/2001]
  Find a right triangle such that the hypotenuse is a square and the sum of the two perpendiculars, or indeed of all three sides, is also a square...

• Exponential Series Proof [Jake, 05/05/2001]
  Given e^x greater than or equal to 1 + x for all real values of x,and that (1+1)(1+(1/2))(1+(1/3))...(1+(1/n)) = n+1, prove that e^(1+(1/2)+ (1/3)+...+(1/n)) is greater than n. Also, find a value of n for which 1=(1/2)+(1/3)+...+(1/n) is greater than 100.

• Last Four Digits of the Fibonacci Numbers [Sam, 05/06/2001]
  Show that there is a number ending with four zeros in the Fibonacci sequence; prove that the Fibonacci sequence has a cycle for the last four digits with a length of 15,000.

• Zero to a Negative Exponent [Bailis, 05/06/2001]
  Is 0^(-3) equal to 0, or is it undefined? We can't determine whether to use the 0^x = 0 rule, or to interpret it as 1/(0^3).

• Sum of Distinct Fibonacci Numbers [Sam, 05/06/2001]
  How do you show that every positive integer is a sum of distinct terms of the Fibonacci sequence?

• Converting any Number Base to Another [Trish, 05/11/2001]
  Is there a particular technique for the conversion of any number base into any other base? I also need information on basic numeric calculations within a base 5 system.

• Stirling's Approximation [Faith, 05/16/2001]
  Is there a way to get the answer to a factorial without having to multiply out all the numbers?

• Find the Smallest Triangle [Malau, 05/25/2001]
  A triangle has sides whose lengths are consecutive integers. Its area is a multiple of 20. Find the smallest triangle that satisfies these conditions.

• Integer Logic Puzzle [Andrew, 04/22/2001]
  Two integers, m and n, each between 2 and 100 inclusive, have been chosen. The product is given to mathematician X and the sum to mathematician Y... find the integers.

• Converting to Binary [Maya, 05/27/2001]
  Converting the number .78125 to its binary equivalent.

• Converting Negative Decimals to Hexadecimal [Josh, 05/30/2001]
  Using a calculator I can get conversions like - 16 = F0, but what I need to know is how to get to F0 from - .16

• Numerically Equal Volumes and Surface Areas [Irene, 06/04/2001]
  Find all rectangular solids with integral dimensions, the volumes and surface areas of which are numerically equal.

• What is Modulus? [Bryan, 06/06/2001]
  I have used the mod command and know what the results mean, but I don't understand the theory behind it and what is actually happening.

• Perimeter of Pascal's Triangle [Bobby, 06/05/2001]
  Is there a general formula for finding the perimeter of Pascal's triangle using the number of rows?

• Stirling Numbers of the Second Kind, Bernoulli Numbers [Mahdi, 05/29/2001]
  Sk = 1^k+2^K+3^k+...+n^k. Find Sk as a formula.

• Second-Order Linear Recurrences [Lanada, 06/08/2001]
  Three problems involving recurrence equations.

• Egyptian Fractions [Andrew, 06/11/2001]
  The Egyptians wrote all their fractions as a sum of different fractions with a numerator of 1. I need to find a way to work out what fractions should be added together...

• Floating 2 [Britney, 06/12/2001]
  Start with the number 2 on the far left side, then float the number 2 to the far right side; the new number must be three as large as the old number.

• Determining Primes by Their Square Roots [Paul, 06/13/2001]
  My problem has to do with determining if a very large number is a prime.

• Fermat's Little Theorem: A Special Case [Saquib, 06/26/2001]
  Show that n^7-n is divisible by 7.

• Formula for the Extraction of a Digit [Jake, 06/18/2001]
  Is there a formula to extract any digit of a given number?

• Non-Periodic, Non-Terminating Decimals [Rebecca, 06/26/2001]
  Why is a non-periodic, non-terminating decimal an irrational number?

• Multiplication of Two Negative Numbers [Daniel, 06/28/2001]
  To what extent do the negative numbers we use deserve to be called numbers?

• Diophantine Equations [Deva, 06/29/2001]
  Find rational x and y such that x^2+x^2*y^2 and y^2+x^2*y^2 are perfect squares, or, more simply, x^2+x^2*y^2 = m^2 and y^2+x^2*y^2 = n^2, where n and m are rational numbers.

• Definition of Floating Point Data [Judy, 07/02/2001]
  What are 'floating point data'? How do they differ from an integer? What are some examples?

• One to the Power of Infinity [Matt, 07/03/2001]
  An instructor says that one to the infinite power does not equal one. If this is true, is there a relatively simple explanation?

• Induction Proof with Inequalities [Jay, 07/03/2001]
  Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative integer.

• Investigation Involving Square Root of 2 [Loka, 07/09/2001]
  How can you explain the fact that (665857/470832)^2 = 2?

• Sum of Unit Fractions [Candyce, 07/17/2001]
  By induction, prove that every proper fraction p/q with p less than q can be written as a finite sum of distinct reciprocals of positive integers.

• Zero and Imaginary Numbers [Angela, 07/18/2001]
  Is it true that zero divided by an imaginary number is zero? How could the answer be in the real number line when the divisor can't be found in the real number line?

• Sum of Integers [Mark, 07/03/2001]
  How many integers are 13 times the sum of their digits?

• Perfect Square [Lory, 07/23/2001]
  If g.c.d.(x, 3) = 1 and g.c.d.(y, 3) = 1, show that x^2 + y^2 cannot be a perfect square.

• Chinese Remainder Theorem and Modular Arithmetic [Donald, 07/21/2001]
  Professor Carroll tries to divide his class into three groups, but two students are left...

• One equals Two [Dick, 07/25/2001]
  There is an algebraic manipulation involving division by zero that results in one equals two, or some other contradiction. What is it?

• Incommensurable Numbers [Fries, 07/25/2001]
  What is an incommensurable number?

• Largest 7-Digit Number [Patrick, 07/27/2001]
  Work out the largest 7-digit number you can applying two rules: every digit in the number must be able to be divided into the number, and no digit can be repeated.

• Sum of Consecutive Odd Integers [Hooji, 07/27/2001]
  Given an integer N, can N can be written as a sum of consecutive odd integers? If so, how can I identify *all* the sets of consecutive odd integers that add up to N?

• Fractions between 0 and 1 [Courtney, 07/29/2001]
  Is there a way to find the number of different (no equivalent fractions) fractions between 0 and 1 with denominators from 2 to 100 without writing out every fraction and counting them?

• Is Zero a Perfect Square? [Patricia, 08/01/2001]
  I have been told that zero is not considered a perfect square, yet the square root of zero is zero...

• Find the Flaw [Dan, 08/02/2001]
  I don't understand where the following proof goes wrong...

• Converting Binary and Decimal to Hexadecimal and Back [Robert, 08/12/2001]
  How do you convert binary and decimal numbers to hexadecimal and vice versa?

• 1997M [Brian, 08/12/2001]
  Find all composite positive integers M such that the product of 1997 and M has exactly four divisors.

• Equations with a Common Root [William, 08/22/2001]
  Find all real numbers a such that the equations x^9+ax^7-(a-3)x^6-1/ 2x^2+1=0 and 2x^5+2ax^3-(2a-6)x^2+1=0 have a common root.

• Subsets of Real Numbers and Infinity [Kevin, 08/22/2001]
  Am I correct in saying that both the whole number set and the integer set have an infinite number of numbers within them, and therefore are of the same size?

• Sum of Powers of 2 [Amir, 08/28/2001]
  I want to derive a formula for the sum of powers of 2.

• Conjecture About Squares of Consecutive Numbers [Bryan, 08/28/2001]
  When any two rational numbers whose absolute values are 1 apart are squared, the difference of the squares is equal to the absolute value of the sum...

• Birthday Calendar Puzzle [Nancy, 08/29/2001]
  My question involves a game that I have played with my students for a long time, yet I am always unable to explain to them why the pattern works...

• Counting Positive Rational Numbers [Alfred, 09/09/2001]
  In Hardy's book _Pure Mathematics_ he gives a formula for counting the positive rational numbers p/q when they are arranged in a triangular matrix and counted down diagonally from the top row... how can it be proved for all such numbers?

• Repeating Decimals - Rational or Irrational? [Mirko, 09/11/2001]
  Are 0.252252225... and 0.125126127... rational or irrational?

• Why Use Q and Z? [Jim, 09/12/2001]
  Why is the letter Q used for rational numbers and Z for integers?

• How Many Rectangular Solids in a Cube? [Right, 09/13/2001]
  Is there any standard way of finding out how many different possible rectangular solids can fit into an 3^3 cube?

• Base of an Exponential Function [Stefanie, 09/15/2001]
  Why can't the base of an exponential function be negative?

• n Factorial - Prove Lower Bound is n^(n/2) [Johnny, 09/11/2001]
  I am trying to prove the following: n^(n/2) <= n!

• Euclid's Extended Algorithm [Anna, 09/16/2001]
  Can you please state for me the steps of Euclid's extended algorithm in simple terms?

• Prove x^2+y^2 Not Divisible by 4 [Robert, 09/20/2001]
  Prove that if x and y are odd, then x^2 + y^2 is even but not divisible by 4.

• Proof with Pigeonhole Principle [Matt, 09/20/2001]
  Prove that among five points selected inside an equilateral triangle with sides of length 2, there always exists a pair at a distance not greater than 1.

• Density Property of Rational Numbers [Benjamin, 09/21/2001]
  How is the density property of rational numbers proven?

• Irrationality of e+pi and e*pi [Andy, 09/24/2001]
  I have read that it is unknown whether either E+Pi or E*Pi is an irrational number. How can we prove that at most one of the two numbers is rational?

• Binary Addition (2s Complement) [Paul, 09/25/2001]
  Given a number in binary (10110111), I have to convert it to decimal using 2s complement, then to BCD (binary coded decimal). How does 2s complement work?

• Given Irrational Numbers a,b, Is a^b Rational? [JM, 09/26/2001]
  Is it possible to demonstrate that there are irrational numbers a,b such that a^b is a rational number?

• Find the Smallest Number - A Remainder Problem [Bob, 09/27/2001]
  Find the smallest number, M, such that: M/10 leaves a remainder of 9; M/9 leaves a remainder of 8; M/8 leaves 7; M/7 leaves 6; M/6 leaves 5; M/5 leaves 4; M/4 leaves 3; M/3 leaves 2; and M/2 leaves 1.

• Is Zero a Real Number? [Patrick, 09/27/2001]
  My friend and I think we have disproved that 0 is a real number.

• Complex Powers [Bill, 09/28/2001]
  How do I show that abs(z^i) is less than exp^pi where z is a complex number not equal to 0?

• Irrational Numbers x,y, x^y Rational? [Joe, 09/28/2001]
  Are there any irrational numbers x and y such that x^y is rational?

• Prime Number Proof: p_2n Greater Than 2*p_n [Henry, 10/07/2001]
  Prove that for n greater than 1, p_2n is greater than 2*p_n, where p_n is the nth prime number.

• Reverse Modulus Operator [Charles, 10/09/2001]
  Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

• Sums Divisible by 11 [Chris, 10/10/2001]
  Why is the sum of a number with an even number of digits and that same number written in reverse always divisible by 11?

• Identity Element [Cami, 10/12/2001]
  What is an "identity element"?

• Bar Codes and Check Digits [Melissa, 10/13/2001]
  What do bar codes have to do with math?

• What Exactly is a Fraction? [Sridhar, 10/15/2001]
  What is a fraction? Is 3/1 a fraction? Is 5/sqrt(2) a fraction?

• Rational and Irrational Numbers: Multiplication, Division [Jess, 10/15/2001]
  I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational.

• p, p+8, p+22 Not Prime [Scott, 10/16/2001]
  Prove that there is no positive integer p such that each of the numbers: p, p+8, p+22 is prime.

• Coding Pairs of Numbers [SoonMin, 10/18/2001]
  Using the equation: 1/2 ((a + b)^2 + 3a + b) I have plugged in numbers for a and b and worked it out, but I do not see how that "codes the pair" (a,b) into a single number.

• About Base Five [David, 10/22/2001]
  I don't understand anything about bases.

• Decimal Expansion of a Reciprocal [Gudlaug, 10/23/2001]
  1/(X + Y + Z) = 0.XYZ.

• Math Virus Formula [Victor, 10/23/2001]
  The virus spreads to all the squares directly touching each other (not including diagonally) and I have found the formula for the number of newly infected cells (although this does not include the first minute)...

• Fundamental Theorem of Arithmatic [Josie, 10/23/2001]
  How do I prove that the cube root of 2 is an irrational number using the Fundamental Theorem of Arithmetic?

• No Integer Solution [Tomas, 10/21/2001]
  I don't know how to prove that the following equation has no solution in Z (integers): z^2-2y^2=51.

• Product and Sum of Digits = Number [Meghan, 10/24/2001]
  How many two-digit numbers exist such that when the products of their digits are added to the sums of their digits, the result is equal to the original two-digit number?

• Perfect Square [Robin, 10/26/2001]
  If a and b are positive integers such that (1+ab) divides (a^2+b^2), show that the integer (a^2+b^2)/(1+ab) must be a perfect square.

• GCD Even/Odd Proof [Paras, 10/26/2001]
  If m is greater than nn and a,m,n are positive with m not equal to n, prove that the GCD of (a^2^m+1, a^2^n+1) = 1 if a is even; and 2 if a is odd.

• Inequality Proof for Greatest Integer [Lisa, 10/27/2001]
  If x is an arbitrary real number, prove that there is exactly one integer n that satisfies the inequalities n equal or greater than x less than n+1.

• Summing a Series Like n*(n!) [Aberlig, 10/28/2001]
  How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?

• Base 2, Base 8 Multiplication and Addition [Barbara, 10/28/2001]
  When to carry: multiplying in bases other than 10.

• What is 0^0? [Molly, 11/01/2001]
  We are doing exponents in school and we were talking about how 9^0=1, 10^0=1, etc., and I asked what 0^0 is.

• How Many Triangles? [Aliaa, 10/30/2001]
  If we join one point on each of the three sides of a triangle to make another triangle, there are three triangles with vertices pointing up. How many triangles will have vertices pointing up if there are n points?

• Factorial Base and Base 10 [Leonard, 11/02/2001]
  Let n be a number written in base 10, which also has an interpretation in factorial base. Let m be the value of its interpretation in factorial base. What is the greatest n for which m is equal to or less than n?

• Linear Proof [Emily, 11/07/2001]
  We say that f is linear provided that for every x, y in its domain, f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the set of real numbers), then f is defined by f(x) = cx for some c belong to R.

• Modular Arithmetic [Chaim, 11/08/2001]
  For any integer a, a^4 is congruent to 0 or 1 (mod 5)... We were able to work with the odd number case so it is just the even case that is getting us stuck.

• Degree of Constant Function [Masha, 11/08/2001]
  We think F(x) = 1x^0 is not a polynomial function (because polynomials shouldn't have discontinuities), but F(x) = 1 is a polynomial. And F(x) = 1 still has degree 0 but for reasons we can't explain.

• Polynomial Divisible by 7 [Lindsay, 11/14/2001]
  Prove that 2^(3n+1) + 4^(3n+1) + 1 is divisible by 7.

• Are 0 and 1 Really Rational Numbers? [Richard, 11/14/2001]
  Here's when the laws of rational numbers fall apart: A) 0/1 = 0; B) 0/ 0 = 0 and 1.

• Numbers Raised to the Negative Power [Rosemary, 11/14/2001]
  I know that 5^(-N) = 1/5^N. I would like to know why.

• Odd Primes and Primitive Roots [Alison, 11/14/2001]
  Given distinct odd primes P and Q, prove that N = PQ has no primitive roots.

• Equation without a Solution [Steven, 11/14/2001]
  What is the solution to the equation sqrt(x) = -2 ?

• Does Infinity Exist? [Muhammad, 11/15/2001]
  What proof do we have that infinity actually exists?

• Algebra Tiles and Negatives [David, 11/05/2001]
  Use a model (algebra tiles) to demonstrate that a negative times a negative = a positive.

• Prime and Consecutive Numbers [Debbie, 11/16/2001]
  Why are 3, 5, and 7 the only numbers that appear to be prime and consecutive?

• Non-negative Integers [Jannet, 11/15/2001]
  How many nonnegative integers consisting of 1-3 digits are divisible by 5? How many nonnegative integers consisting of 1-3 different digits are divisible by 5?

• Zeros between 1 and 222 Million [Gerrit, 11/17/2001]
  How many zeros will I use if I write down all the numbers from 1 to 222 million? And how can I generalize this?

• Beyond Pythagorean Triples [Lynn, 11/20/2001]
  a^3+ b^3+c^3 = d^3 and a^4+b^4+c^4 = d^4: equations that are more difficult than the situation with Pythagorean triples.

• Consecutive Fibonacci Numbers Relatively Prime [Beverly, 11/17/2001]
  Prove that two consecutive Fibonacci numbers are relatively prime.

• Finding 13^99 [Potter, 11/21/2001]
  What is the units digit of 13 to the 99th power?

• Taylor Expansion [Henning, 11/21/2001]
  Can you give me the proof of this statement: arcsin(x) = x + 1/2 (x^3/ 3) + (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) + ... The basis of the calculation is a Taylor series.

• Prime Number Formula [Ri, 11/11/2001]
  What formula did Leonhard Euler use to find prime numbers?

• Squares in an Infinite Factorial Series [Gaurav, 11/23/2001]
  How many perfect squares appear among the following numbers: 1!, 1!+ 2!,1!+2!+3!,...1!+2!+3!+...n!?

• Negative Numbers to Powers [Filoxus, 11/23/2001]
  If u is an irrational number and x is a negative number, what is x^u? How do I determine even whether that number is positive or negative?

• Primality Test [Maria, 11/26/2001]
  I want to write a program using Pascal that will verify whether a number is prime.

• Unknown Numbers and a Venn Diagram [Christina, 11/26/2001]
  The GCF of two numbers is 20 and the LCM is 840. One of the numbers is 120. Explain how to find the other number and use the Venn diagram method to illustrate.

• Binet's Formula and Induction [JT, 11/28/2001]
  What is induction, and can you prove Binet's formula by induction?

• Fibonacci Sequence Property [Enrique, 11/29/2001]
  I have to prove that in the Fibonacci sequence, F(k) is a divisor of F(nk), where n is a natural number (so, F(nk) = A*F(k) where A is a natural number).

• What is a Property? [Jim, 11/29/2001]
  I understand Undefined and Defined terms and Axioms and Theorems, but what exactly is a Property? Is it the same thing as a Theorem? Also what is a Law?

• When Casting Out Nines Fails [Teresa, 11/29/2001]
  To prove that casting out 9's worked, I intentionally used an incorrect quotient, but was shocked when the problem checked out correctly.

• House of Cards [Jo, 12/02/2001]
  Is there a rule for working out the number of cards you need to build a house of cards of any size?

• Proof Regarding LCM [James, 12/05/2001]
  Is there a proof of the equation: given integers a and b, a*b = GCF(a,b) * LCM(a,b)?

• What Kind of Number is One? [Reed, 12/06/2001]
  What is one, if it is neither prime nor composite?

• Prime Triplet [Lynn, 12/07/2001]
  The consecutive odd numbers 3,5,7 are all primes. Are there infinitely many such 'prime triplets'?

• Fibonacci Identity [Doug, 12/10/2001]
  I am trying to create an inductive proof for the particular identity of Fibonacci numbers that: F(n-1) * F(n+1) = (-1)^n + (Fn)^2.

• History of Properties [Noah, 12/11/2001]
  Who invented the properties? For example, the distributive property and the associative property.

• Irrational Pi [Zach, 12/22/2001]
  Why must pi be irrational?

• Finding Sets of Integers [Keith, 01/02/2002]
  Without computer assistance, find five different sets of three positive integers such that and k is less than and m is less than n, and 1/k + 1/n + 1/m = 19/84.

• Largest Integer Divisible by All Integers [Salman, 01/01/2002]
  Show that 24 is the largest integer divisible by all integers less than its square root.

• Prove a and b are Perfect Squares [Salman, 12/28/2001]
  Let a and b be positive integers such that (a,b) = 1 and ab is a perfect square. Prove that a and b are perfect squares.

• A Solution in Natural Numbers [Salman, 10/30/2001]
  Prove that x^2+y^2=z^n has a solution in natural numbers for all n, where n is a natural number.

• Sum of Two Cubes [Gemm, 01/12/2002]
  Find the smallest number that can be expressed as the sum of two cube numbers in two different ways.

• Primes Greater Than/Less Than Multiples of Six [Riley, 01/18/2002]
  Has the postulate stating that every prime number is either one more or one less than a multiple of six, excluding 2 and 3, been proven?

• 0 As the Denominator [Kyle, 01/19/2002]
  Can zero over zero equal anything?

• Perfect Squares: n+125 and n+201 [Lisa, 01/21/2002]
  Find the smallest positive integer n so that n+125 and n+201 are both perfect squares.

• Can a Number to the Zero Power be -1? [Cheryl, 01/25/2002]
  My teacher told me that (-3^0) is equal to -1. Is this true?

• Prove a = b = c [Jim, 01/27/2002]
  When a^2 + b^2 + c^2 = ab + bc + ca and abc does not equal 0, prove that a = b = c.

• Perfect Square, Cube, Fourth Power [James, 01/25/2002]
  Find the least integer greater than 1 that is a perfect square, a perfect cube, and a perfect fourth power.

• Mathematical Induction [Ali, 01/28/2002]
  Use Mathemetical Induction to prove that any postage of at least 8 cents can be obtained using 3- and 5-cent stamps.

• ((n+1)/2)n [Alan, 01/31/2002]
  If you want to figure out the total of a series of numbers in order, e.g.: 1+2+3+4+5+6+7+8+9 etc., you would use the formula ((n+1)/2)n, where n is the final number of your series. Why?

• Converting a Base 2 Log [Sue, 01/31/2002]
  2147483647 is one less than what power of 2?

• Even and Odd Numbers in Base 5 [Sheri, 02/02/2002]
  How can you tell if a number in base 5 is even or odd?

• Convergence of a Prime Sequence [Ken, 02/10/2002]
  We know the sum(1/n) (n=1, infinity) does not converge, but what about sum(1/p) where sum is over prime numbers only. Does it converge?

• Generalised 'Fibonacci' Series and Phi [Stuart, 02/10/2002]
  A Fibonacci-style series that starts with any two numbers and adds successive items produces a ratio of successive items that converges to phi in about the same number of terms as for the basic Fibonacci series. Is this well known and provable?

• Converting from Base 10 to Base 3 [Ian, 02/12/2002]
  How do you convert a base 10 number (example 2315) to base 3?

• Divisibility Proof for Odd Integers [Mike, 02/13/2002]
  Prove that for all odd integers N, N^3 - N is divisible by 8.

• Perfect Squares and Irrational Numbers [Mimi, 02/13/2002]
  Isn't any non-perfect square an irrational number? What is the number 0.49? Its square root is 0.7, which is neither irrational nor an integer.

• Divisibility by 11: Proof [Jenny, 02/12/2002]
  Prove that a positive integer n is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

• Finding Mersenne Primes [Hanif, 02/15/2002]
  How do I find the first four Mersenne primes?

• Perfect Squares with Congruences [Stacey, 02/16/2002]
  Prove that there is no perfect square a^2 whose last digits are 35.

• Undefined Fractions [Sandy, 02/19/2002]
  Why is a fraction with a denominator of zero called "undefined"?

• Perfect Square? [Charles, 02/18/2002]
  If we use the digits 1,2,3,4,5,6,7 each only once to form a 7-digit number, can the resulting number be a perfect square?

• If n^2 is Even, n is Even [Mary, 02/21/2002]
  I have to show that if n^2 is even then n is also even.

• Double Factorial [Linda, 02/22/2002]
  Can you tell me what two ! marks mean in factorial questions?

• Accountants Use 9 to Check for Errors [Steven, 02/21/2002]
  I know accountants divide the difference of debits and credits by 9 to check for a transposition error. I need to understand why this works.

• Sum of Two Different Primes [Nate, 02/22/2002]
  Can the sum of two different primes ever be a factor of the product of those primes?

• Inconstructible Regular Polygon [Roger, 02/22/2002]
  I've been trying to find a proof that a regular polygon with n sides is inconstructible if n is not a Fermat prime number.

• Division by Zero: Indeterminate or Undefined? [Brant, 02/23/2002]
  I'm having some trouble understanding division by zero.

• Binary Division and Negative Binary Numbers [Marc, 02/24/2002]
  Suppose you enter 11111000 into a binary calculator. How would it know whether that is to be 248 or -8? How does the computer begin the division process? What does it do when it cannot right shift the bits?

• Euler Phi Function [Rachel, 02/24/2002]
  If p and q are prime, investigate: phi(p^n * q^m).

• Perfect Square Equation [Barry, 02/22/2002]
  Prove that if n is greater than 1, then nC2 + (n-1)C2 is a perfect square.

• Converting Fractions from Binary to Decimal [Jay, 02/25/2002]
  Can you explain how to convert binary fractions to decimal numbers, e.g. 0.00011001100110011001...?

• Product of Primes [Ganesh, 02/27/2002]
  Can you provide me with the proof that every non-zero positive integer can be written as a product of primes?

• Multiplying by Zero [Jess, 03/01/2002]
  Why does any number times zero equal zero?

• Numbers in the Fibonacci Sequence [Dinu, 07/19/2001]
  How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ?

• Wilson's Theorem [Hanneke, 03/03/2002]
  I'm looking for a proof for Wilson's theorem: n divides (n-1)! + 1 if and only if n is a prime number.

• Pascal's Triangle and Fibonacci Formula [Yael, 02/23/2002]
  Prove that the diagonals of Pascal's triangle are the Fibonacci numbers.

• Proof by Contraposition [Adam, 03/06/2002]
  How can I prove that n^6 + 2n^5 - n^2 - 2n is divisible by 120?

• Base e, In, Log [Jerry, 03/08/2002]
  Problems in base e. For example: 3e^(2x-1) = 7, or e^(x+1) = 8.

• Large Numbers and Congruences [Boris, 03/05/2002]
  Find the last three digits of the number 11^(11^(11^(11^(11^11)))) written in base seven.

• Counting Digits [Susan, 03/13/2002]
  How many times does each digit appear when counting from 1-1000 and 1- 10000 (including zeros)?

• Was Euler wrong? 2*Pi=0? [Warren, 03/13/2002]
  While I was surfing the Internet, I found a site with an interesting proof that shows that 2*Pi = 0 by using Euler's famous equation...

• Product Always an Even Number? [Tian, 03/17/2002]
  The letters a1, a2, a3, a4, a5, a6, a7 represent seven positive whole numbers; b1, b2, b3, b4, b5, b6, b7 represent the same numbers but in a different order. Will the value of the product (a1-b1)(a2-b2)(a3- b3)(a4-b4)(a5-b5)(a6-b6)(a7-b7) always be an even number?

• TI-86 Base Conversion Program [Amber, 03/19/2002]
  I have finished writing a program that can convert any number in any base (one-ten) to base ten. Now I am writing a program to convert any number in base ten to a given base.

• Fraction Algorithm [Gil, 03/19/2002]
  I have been having trouble making an application that can convert a finite decimal to a fraction without doing 78349/1000000.

• e^pi vs. pi^e [Rajesh, 03/20/2002]
  Which is greater, e^pi or pi^e? I would like to have a simple proof.

• Products of Integers (Even or Odd) [Kristine, 03/23/2002]
  How can I prove that the product of two even integers is an even integer and the product of two odd integers is an odd integer?

• Integers and Fractions [Filgi, 03/23/2002]
  Give an example of a positive integers p,a,b where p/ab and not p/a and not p/b. Let m, n, and c be integers. Show that if c/m then c/mn.

• Prime Integer Proof [Shane, 03/24/2002]
  Prove that if p is a prime number greater than or equal to 5, then there exists an integer k such that p=sqrt(24k+1).

• Binary Subtraction [Robert, 03/25/2002]
  I keep getting lost when doing the following: 1011000-110010.

• Applications of Different Bases [LM, 03/27/2002]
  I am trying to find three bases other than base 2, and find a use for them.

• Finding Products of a Range of Numbers [M, 03/28/2002]
  What method or formula is there to solve for the product of a range of numbers?