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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 COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Number Theory Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Testing for primality. Finding the Two Squares [06/11/2003] One of Fermat's theorems says that every prime number that yields a remainder of 1 when divided by 4 can be expressed as the sum of two integer squares (e.g.: 97 = 4^2 + 9^2). This theorem was proven by Fermat. What methods are known for determining the two squares? Find Remainders: 3^2002/26, 5^2002/26 [09/21/2002] Find the remainders obtained when 3^2002 and 5^2002 are divided by 26; show that 3^2002 + 5^2002 is divisible by 26. Find the Flaw [08/02/2001] I don't understand where the following proof goes wrong... Find the Smallest Triangle [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. Find the Solution: r^2 + s^2 = c. [01/28/2003] Given c, find a^2 + b^2 = c^2. Finite Series and Greatest Integers [03/06/2003] For n a positive integer, let t(n) denote the number of positive divisors of n (including n and 1), and let s(n) denote the sum of these divisors. Prove the following:... Formula for phi(n) [05/09/2003] Find a formula for phi(n) where n is any positive integer. 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 number. 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. 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? 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? Genus of a Plane Curve [05/04/2007] How one can determine the genus of a given curve F(X,Y) in Z[X,Y]? Gosper's Verstion of Stirling's Formula [05/30/2002] I came across a formula by Gosper which seems to be an improvement on Stirling's formula. Can you show me how to derive this formula? 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 colour... Greatest Common Factor [03/28/1997] How do you find the greatest common factor? Greatest Common Factor of Numbers Composed of All Ones [06/30/2008] A quick proof of why any two numbers composed entirely of ones, with one number having one more, such as 1111 and 11111, are relatively prime. 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? Hill Cypher [03/19/2001] What is the Hill Cypher? Can I decode the Hill Cypher by finding the inverse of a matrix with all its elements in mod n arithmetic? How Many Digits Are in the Root? [07/03/2008] What is a method for finding the number of digits in the square root of a 29-digit number? How about any root of any number with a given number of digits? How Many Digits in Graham's Number? [11/11/2005] I have heard that Graham's number is the largest number with mathematical use. I have seen it expressed in arrow notation but that does not give me a sense of how large it is. Is there a way to express the number of digits it contains? Idempotents of Z(n) [10/10/2000] What are the idempotents of Z(n) when n is twice a prime? Inconstructible Regular Polygon [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. Induction Proof with Inequalities [07/03/2001] Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative integer. An Inductive Proof [02/13/2003] If gcd(n,m) = 1, prove gcd(Rn,Rm) = 1. Inductive Proof of Divisibility [06/25/2002] How do you prove that for any integer n the number (n^5)-n is divisible by 30? Infinite Continued Fraction [05/15/2002] What can you determine about the value of the infinite continued fraction [1;1,2,3,1,2,3,1,2,3....]? Integer Iteration Function [12/24/2003] Let X be a positive integer, A be the number of even digits in that integer, B be the number of odd digits and C be the number of total digits. We create the new integer ABC and then we apply that process repeatedly. We will eventually get the number 123! How can we prove that? Integer Logic Puzzle [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. Integer Proof Using Diophantine Equation [01/10/2005] How do you prove that the integer 26 is the only integer preceded by a a square (25) and followed by a cube (27)? Integer Solutions of ax + by = c [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? Integer Solutions to 9r^3 - t^3 - s^3 = 6rst [04/25/2013] A number theorist exchanges observations with Doctor Vogler, touching on elliptic curves, Cardano's method for solving cubics, and Fermat's Last Theorem for small odd powers. Integer Solutions to a^2 - b^2 = k for a Given Integer k [05/07/2005] How to find integer solutions (a,b) to the equation a^2 - b^2 = k for a given integer k. Integer Solutions to a Cubic Equation [04/11/2005] Fermat's method of infinite descent is used to show that the cubic equation (a^3) + (2b^3) + (4c^3) - 4abc = 0, with a, b, and c whole numbers and without a=b=c=0, has no solution. Interesting Diophantine Equation [12/06/2005] Find all integers x such that x^2 + 3^x is the square of an integer. Interesting Integer Problem with Diophantine Equations [06/21/2005] Two positive integers are such that the difference of their squares is a cube and the difference of their cubes is a square. Find the smallest possible pair and a general solution for all pairs (a,b) that satisfy the statement. An Introduction to Basic Diophantine Equations [08/27/2007] A birdcage contains both 2-legged and 1-legged birds, and there are a total of 11 legs in the cage. Use a Diophantine equation to find all possible combinations of birds. Page: []

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