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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'? Pythagorean Triples [05/22/1999] What is the general formula for all sides of any triple? Pythagorean Triples [05/31/1999] Is there a procedure for finding Pythagorean triples? Pythagorean Triples [5/18/1995] How can the relation between Pythagorean triples be expressed as a formula? Pythagorean Triples Divisible by 5 [11/17/2000] Do all right triangles with integer side lengths have a side with a length divisible by 5? Pythagorean Triple with 71 [12/07/1997] Is there a Pythagorean triple that contains the number 71? Quadratic Residues [06/30/1998] I need a fundamental explanation of the concept of quadratic residues. Quadratic Residues and Sums of Squares [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... A Quartic Diophantine Equation: 10657 + 11579x^2 + x^4 = y^2 [12/29/2008] Doctor Vogler helps a student look for integer solutions to a quartic polynomial by noticing a difference of squares in its coefficients and factoring its constant term. Ramsey's Theorem and Infinite Sequence [06/01/1999] Ramsey's Theorem applied to divisibility in infinite sequences. Rational and Irrational Numbers: Multiplication, Division [10/15/2001] I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational. Rationalizing a Denominator with Multiple Cube Roots [04/22/2011] A student of field theory wonders how to remove the cube roots from the denominator of 1/(a + b*CBRT(q) + c*CBRT(q)^2). Building on the conjugacy of square roots, Doctor Vogler writes out the required conjugates. Real and Rational Numbers [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...)? Real Numbers [08/08/1997] What exactly is a real number? Reasoning about Integers [10/06/2004] When positive integers p and q are divided by an even positive integer t, they have remainders 2 and t/2, respectively. What is the remainder when the product pq is divided by t? Reciprocals of Integers Greater Than 1 as Sum of a Series [07/01/2004] Show that the reciprocal of every integer greater than 1 is the sum of a finite number of consecutive terms of the series 1/[j(j + 1)]. Rectangular Solids from Blocks [09/25/1998] How many rectangular solids can be made from "n" cube-shaped blocks? Recurrence Relation for a Pell Equation [11/09/1999] Can you help me find a recurrence relation for generating solutions to the Pell equation x^2 - 5y^2 = 1? Relationship Between GCF and LCM [05/22/2002] What is the exact relationship between the gcf or gcd and the lcm of two numbers? Relatively Prime [10/07/1999] What does the term relatively prime mean, and how can you determine if two numbers are relative primes? Relatively Prime Pythagorean Triples [09/13/1997] Questions about Pythagorean triples. Remainders, Pigeons, and Pigeonholes [03/26/2003] Given 17 integers, prove that it is always possible to select 5 of the 17 whose sum is divisible by 5. Remainder when Dividing Large Numbers [04/17/2001] How can I find the remainder when (12371^56 + 34)^28 is divided by 111? Repeating Decimals [04/28/1999] I am interested in finding longer repeating groups in number tails of repeating decimals. Repeating Decimals - Rational or Irrational? [09/11/2001] Are 0.252252225... and 0.125126127... rational or irrational? Repeating Digits of Fractions [04/28/1999] Do you know any theorems relating to the length of the repeating portion of the decimal representation of fractions? Reversal of Age Digits Every Eleven Years [11/06/2007] Every 11 years, my age is the exact reverse of my mother's age. When I was 13 she was 31, when I was 24 she was 42, and so on. Why does this work? Reversed Digits Theorem [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? Reverse Modulus Operator [10/09/2001] Is there an operator that would return 2 when we we do 6 * 0, * being this new operator? Roots of ax^f2 bx+c = 0 [05/22/1997] Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0 are not rational. Rounding Negative Numbers [05/23/2007] I was wondering what -0.5 is rounded to the nearest integer? RSA Encryption [04/25/2002] Decrypt the encrypted message in ciphertext C to find the original plaintext, a string of English letters. Running Time of an Insertion Sort [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? Search for the Largest Prime [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? Second-Degree Two-Variable Diophantine Equation [04/12/2001] Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k. Second-Order Linear Recurrences [06/08/2001] Three problems involving recurrence equations. Second Order Recurrence with Non-Constant Coefficients [05/27/2005] I'm trying to find a closed form solution of a second order recurrence relation with no constant coefficients, specifically: u(n+2) = 2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help? Sequence of Integers [08/12/2008] Find all functions f such that for each n in Z+ we have f(n) > 1 and f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18. Set Theory and GCD and Divisibility [01/27/2003] If 1 <= a <= n and 1 <= b <= n and ab <= n, and if a divides n and b divides n, does that mean that ab divides n given that GCD(a,b) = 1 ? Show 2^(N-1) Congruent to 1(mod N) [02/25/2003] I need to show that if N = 2^p - 1, p prime, then 2^(N-1) is congruent to 1(mod N). Showing a Diophantine Equation Has No Solutions [07/30/2008] Do there exist positive integers m and n such that m^3 = 3n^2 + 3n + 7? Page: [<]

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