See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Formula for Sums of Counting Numbers [09/19/2002]
Use the formula for the sum of the first n counting numbers to find 9+
- Formula for the Extraction of a Digit [06/18/2001]
Is there a formula to extract any digit of a given number?
- Formula for the First Day of a Year [03/18/1998]
Is there an equation to find the first day of a year given the year?
- Formulas for Primes [09/09/2002]
Prove that n^2 + n + 41 does not always produce a prime number for any
whole number n. Explain why n^2 + 8n + 15 never produces a prime
- Four-Digit Palindromes [10/21/1998]
Why is every four-digit palindrome divisible by 11?
- Four-digit Palindromes Divisible by 11 [02/10/1999]
Why are four-digit palindromes divisible by 11?
- Four Positive Integers, Any 3 Sum to a Square [10/06/2002]
Find four distinct positive integers, a, b, c, and d, such that each
of the four sums a+b+c, a+b+d, a+c+d, and b+c+d is the square of an
integer. Show that infinitely many quadruples (a,b,c,d) with this
property can be created.
- Four Variable Diophantine Expression [05/10/2008]
For what pairs of different positive integers is the value a/(a+1) +
b/(b+1) + c/(c+1) + d/(d+1) an integer? How would I solve it?
- Fraction Algorithm [03/19/2002]
I have been having trouble making an application that can convert a
finite decimal to a fraction without doing 78349/1000000.
- Fractions between 0 and 1 [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?
- Frequency of Digits in Pi [04/05/2001]
What digit occurs least frequently in pi?
- From Reduction to Induction [02/03/2011]
Replace any two numbers x and y from (1, 2, ..., n) with the new single quantity x + y
+ xy; continue in this way until only one number remains. To find a formula for the
smallest number possible from this procedure, Doctor Jacques lays the groundwork
for a proof by induction.
- Fundamental Theorem of Algebra [01/25/2001]
What exactly is the Fundamental Theorem of Algebra?
- Fundamental Theorem of Arithmetic [07/08/1997]
What's so fundamental about the fundamental theorem of arithmetic?
- Fundamental Theorem of Arithmetic [10/23/2001]
How do I prove that the cube root of 2 is an irrational number using the
Fundamental Theorem of Arithmetic?
- Fundamental Theorems [10/02/2000]
What are the fundamental theorems of algebra and arithmetic?
- Gaussian Integers [05/12/1999]
Are all real prime numbers also Gaussian prime numbers?
- GCD Even/Odd Proof [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
- General Formula to Find Prime Numbers? [10/12/2005]
I was wondering if it is possible that there exists a general formula to know what numbers are prime, or has it been proven that no such formula could exist? What evidence do we have for either case?
- Generalised 'Fibonacci' Series and Phi [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?
- Generalized Definition of Prime Numbers [02/01/2001]
Examining an extended definition of a prime number.
- Generalized Gauss Sum [07/18/2003]
Is there a formula that computes the sum of numbers from m to n, where
m and n could be positive, negative, or zero?
- General Observation on Prime Numbers [09/03/2004]
Is it true that all prime numbers greater than 5 are of the form 6n +
1 or 6n - 1? I read this on a website, but it's hard to believe.
- Generating Function of Catalan Numbers [04/04/2000]
Can you explain the recurrence relation for the Catalan numbers?
- Generating Pythagorean Triples [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
- Getting 0.99999... [04/15/1998]
Is there any mathematical way to get 0.99999999999......?
- Given Irrational Numbers a,b, Is a^b Rational? [09/26/2001]
Is it possible to demonstrate that there are irrational numbers a,b such
that a^b is a rational number?
- The Golden Ratio [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.
- Graphing y = (-2)^x [05/27/2005]
How do I graph y = (-2)^x, or any function where you have a negative
number raised to a power?
- Graph of y = (-n)^x [01/17/2005]
I am curious as to what the graph of y = (-n)^(x) would look like,
such as y = (-2)^x. My graphing calculator will not show the graph as
anything, but displays many real values in the table of values.
- Graphs - Proving the Infinite Ramsey Theory [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
- Greatest Common Factor [03/28/1997]
How do you find the greatest common factor?
- Greatest Impossible Score in a Game [01/26/2003]
If the two values possible in a game are p and q, the greatest
impossible score is (pq - p - q). Why is it that?
- Greatest Integer Equation [08/06/2003]
I am trying to correctly interpret [[x]]^2 + [[y]]^2 = 1, where f(x)=
[[x]], is the Greatest Integer function.
- Group Sizes and Remainders [05/19/2002]
A farmer can divide his sheep into equal groups of 17; but for any
group size less than 17, he gets a remainder of one less than the
group size. How many sheep does he have?
- Guessing a Mystery Number [05/23/2007]
One person thinks of a natural number and the other person has to try
to guess it by asking questions. The one who knows the number will lie
once at some point, but otherwise be truthful in answering the
questions. What's the best algorithm for finding the number, and
what's the fewest guesses it would take?
- Happy Numbers [06/21/1998]
What are happy numbers?
- Happy Numbers [01/05/2006]
What is a "happy" number? What are all the two-digit "happy" numbers
between one and one hundred that are multiples of four?
- Help with Proofs [10/16/1997]
Could you please help me with some proofs from a course called
- Hexadecimal Subtraction and Multiplication [04/06/2000]
How do you subtract or multiply hexadecimal (base 16) numbers?