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 TOPICS This page:   number theory    Search   Dr. Math See also the Dr. Math FAQ:   0.9999 = 1   0 to 0 power   n to 0 power   0! = 1   dividing by 0   number bases Internet Library:   number theory HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Number Theory Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Diophantine equations.     Infinite number of primes?     Testing for primality.     What is 'mod'? Pairs of Integers [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. Paradox in the Zero Power [09/26/2000] Why does the product of multiplying a number by itself zero times equal 1? Partitioning an Integer [11/14/1998] How many different ways are there of making a number by adding different combinations of three numbers? Partitions and Products [01/02/2003] What is the best way of dividing an integer into parts so that the product of the parts will be as large as possible? Is there a universal law that tells us what partition will produce the maximum product for any given number? And can such a law be proved? Pascal's Triangle and Fibonacci Formula [02/23/2002] Prove that the diagonals of Pascal's triangle are the Fibonacci numbers. Paths to Triangle Points [05/26/1999] How can I find the number of paths to a point using Pascal's triangle? Pattern in Period [01/31/1999] Is there any pattern in odd periods with 1 as numerator? Patterns in Repeating Decimals [08/06/2003] Why do certain number sequences repeat in the decimal expansions of common fractions? Patterns in Rolling 3 Dice [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... Pell Pairs of Positive Integer Solutions [03/29/2012] Solving a Pell equation confuses a student. Doctor Vogler applies modular arithmetic to show the way, then concludes with a survey of online resources into Diophantine equations. Percentage of Numbers Divisible by 6, 8 [03/18/2003] What percentage of numbers is divisible by 6? by 8? Percentages of Prime Numbers [01/31/1999] Does the percentage of prime numbers at every power of 10 decrease until it reaches a constant value? Perfect Logarithms [06/24/2002] What can you tell me about the equation log(abc)= log(a+b+c)? Perfect Number [7/21/1996] How can I find a perfect number? What are some reference books about number theory? Perfect Number Algorithms [03/31/1998] What is the formula for a computer program that tests whether an integer is a perfect integer? Perfect Numbers [11/10/1997] Do the sums of the digits of perfect numbers always equal 1? Perfect Numbers [06/15/2002] Please show that any even perfect number ends in 6 or 8. Perfect Square [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. Perfect Square [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. Perfect Square? [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? Perfect Square, Cube, Fourth Power [01/25/2002] Find the least integer greater than 1 that is a perfect square, a perfect cube, and a perfect fourth power. Perfect Square Equation [02/22/2002] Prove that if n is greater than 1, then nC2 + (n-1)C2 is a perfect square. Perfect Squares and Irrational Numbers [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. Perfect Squares: n+125 and n+201 [01/21/2002] Find the smallest positive integer n so that n+125 and n+201 are both perfect squares. Perfect Squares with Congruences [02/16/2002] Prove that there is no perfect square a^2 whose last digits are 35. Perimeter of Pascal's Triangle [06/05/2001] Is there a general formula for finding the perimeter of Pascal's triangle using the number of rows? The Phi Function [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)? Polynomial Divisible by 7 [11/14/2001] Prove that 2^(3n+1) + 4^(3n+1) + 1 is divisible by 7. A Polynomial in Three Variables with Few Integer Solutions [03/12/2011] A student seeks proof that a polynomial in n and two other variables has no integer solutions. After a little insight from modular arithmetic and a lot of searching with a computer algebra system, Doctor Vogler turns up many solutions. Positives and Negatives with Infinity and Zero [10/05/1999] Are there such things in math as +0, -0, and unsigned 0; and +infinity, - infinity, and unsigned infinity? Are these different? Powers of 2 Proof [03/24/2003] Prove that any number that is not a power of 2 can be expressed as a sum of two or more consecutive positive integers, but that this is not possible for powers of 2. p, p+8, p+22 Not Prime [10/16/2001] Prove that there is no positive integer p such that each of the numbers: p, p+8, p+22 is prime. Prefix for 10^30 Bytes [05/25/2000] What do you call 1,000,000,000,000,000,000,000,000,000,000 or 10^30 bytes? Primality Test [11/26/2001] I want to write a program using Pascal that will verify whether a number is prime. Primality Testing [04/22/1998] Is there any fomula to find if a number is a prime? Primality Testing [12/02/2004] How can I determine if a given number is prime? Prime and Consecutive Numbers [11/16/2001] Why are 3, 5, and 7 the only numbers that appear to be prime and consecutive? Prime Factor [02/18/2002] I need to prove that each integer of the form 3n + 2 has a prime factor of this form. Prime Factors, Modular Arithmetic, and Using Pari [08/08/2007] Suppose we have two positive integers 'a' and 'b'. Is there a method to find a positive integer 'k', such that a + bk = x^2 for some integer 'x'? In other words, how can we find a positive integer 'k', such that 'a + bk' is a square? Prime Factors of 4,194,305 [09/20/1999] How can I find the prime factors of 2^22+1? 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