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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. Congruence Class of 10^n Modulo 11 [04/14/2003] What is the congruence class of 10^n modulo 11? Use this to determine the remainder when 654321 is divided by 11. Conjectures vs. Hypotheses [01/12/1999] What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture? Consecutive Composite Numbers [07/05/2002] Find 1000 consecutive composite numbers. Consecutive Composite Numbers [06/07/2005] Is it possible to find m consecutive composite numbers where m is any natural number? Consecutive Integer Proof [09/16/2004] Is it possible to prove that the product of five consecutive integers cannot be a perfect square? Counting Infinities [04/29/1997] I think I have found a flaw in the uncountability proof. Is this possible? Counting Unique Rational Numbers [11/16/2008] How many unique simple forms of rational numbers are there of the form p/q, where p and q are non-zero whole numbers less than or equal to n? For example, 1/2 and 2/4 have the same simple form, so they are not considered unique. The answer should be a function of n. Cubes as Differences of Squares [07/04/2002] Prove that the cube of any positive integer is equal to the difference of the squares of two integers. Cubic Diophantine Equation in Three Variables [09/10/2004] Find integer solutions of the equation x^3 + y^3 = 31z^3. I know the fundamental solution is (137, -65, 42), but I want to have all the values positive. I know also that there is an arithmetic procedure (doubling in the group) to obtain further solutions from the fundamental one, but I do not know the details of this procedure. Cyclic Redundancy Check [06/26/2002] I understand how cyclic redundancy checks work, but I fail to see how appending zeros to the message string (before the division) provides an advantage. Decomposition of Primes as the Sum of Two Squares [12/17/2005] If you write a prime number p as a sum of two squares, can you prove that its decomposition a^2 + b^2 is unique for (a,b up to +/- 1)? Degree of Error in pi(x) Approximation [10/20/2004] How does the error in the formula x/ln(x), used to approximate pi(x) (the primes counting function), behave for large values of x? Determine b if (a+b)/5 = (b-1)/2 [07/25/2003] Both a and b are 4 four-digit numbers, a is greater than b, and one number is obtained from the other by reversing the digits. Determine b if (a+b)/5 = (b-1)/2. Difference of Square Numbers [07/18/2008] Can one number ever be represented as two distinct differences of squares? Or is every difference of square numbers unique? A Diophantine Determined with a Data-Driven Approach [06/24/2012] A student wonders if a u-substitution would solve a troublesome Diophantine equation. Doctor Vogler takes up the struggle with a false start of his own, before noticing helpful patterns from a computational attack and walking the student through the thought process behind his strategy. Diophantine Equation [08/25/2007] Find all integer solutions (a,b) such that (1-ab-a-b)/(1-ab+a+b) is an integer. Diophantine Equations in Three Variables [10/30/2004] I need to know how to get positive integer solutions of two Diophantine equations having three variables. For example: 2x + 3y + 7z = 32 ; 3x + 4y - z = 19. (Give the positive set of triples for the above equations.) Diophantine Equation Solved by Reasoning [04/05/2006] Find all integer solutions of (1/m) + (1/n) - (1/mn^2) = 3/4. Diophantine Equations, Step by Step [10/01/2002] Find all positive integer solutions to 43x + 7y + 17z = 400. Diophantine Equation with Three Unknowns [09/28/2009] Find all triples of positive integers (x,y,z) such that xyz + xy + yz + zx + x + y + z = 243. Diophantine Equation with Three Variables [02/01/2006] Find all positive integer solutions (a,b,c) that satisfy the equation a + b + c + ab + bc + ca = abc + 1. Diophantine System of Equations [12/31/2004] I'm trying to find the rational solutions (in parametric form) to the set of two simultaneous equations: x^2 + x = y^2 x^2 + 1 = z^2 Direct Conversion of Binary to Octal [05/14/2002] How can you convert from base 2 to base 8 without going through base 10? Discrete Logarithm Problem [10/13/2004] Given a === b^c mod N. When a, b, and N are given, can we find c? Distances between Rational and Irrational Numbers [01/13/2005] Is it true that if |x - p/q| < |x - r/s| with x irrational and p/q, r/s irreducible rationals then q > s? If so, how can you prove it? Divisibility by 8 [04/14/1997] Show that, if n is a positive integer, then 5^n+2*3^(n-1) + 1 is divisible by 8. Divisibility Proof [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? Divisibility Proof by Cases [02/23/2003] Prove that if d|n, then (2^d - 1)|(2^n - 1). Do Rational and Irrational Numbers Alternate? [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? Elliptic Curve Factorization [02/10/2006] I would like to find out how to develop the parameters of a cubic parabola in general so that I can implement an integer factorization method. I would also like to know how to add points such as P+P and P+Q to such a curve. Elliptic Curve Resources [05/04/2007] Is there an algorithm to determine the rank of a general elliptic curve? What are some good books to start learning about elliptic curves? Equation with Two Exponential Terms [06/27/2009] Find all ordered pairs (a,b) for which 3^a + 7^b is a perfect square. Erdos' Proof [04/03/1997] Can you show me Erdos' proof that there is a prime number between every integer n and 2n? Euclidean Algorithm [5/13/1996] How can we prove that Euclid's method for finding the highest common factor for two numbers will work for all values? Euclidean Algorithm and Linear Equations [11/03/2003] Could you please explain step by step how to use the Euclidean Algorithm to solve a linear equation and find x and y integers? Euclidean Algorithms [3/13/1996] What is the Euclidean algorithm? What is a "constructible" number? What can you tell me about Diophantine equations? Euclidean and Division Algorithms [11/26/1997] Can you show and explain the proofs of the Euclidean Algorithm and the Division Algorithm? Euclid's Extended Algorithm [09/16/2001] Can you please state for me the steps of Euclid's extended algorithm in simple terms? Euler's Phi Function Applied to Large Numbers [08/23/2004] I'm trying to find the phi value of a large (10-digit) composite number. Can it be done in polynomial time, and if so, is there an algorithm? Expansion of n(n+1)(n+2)...(n+k) [07/15/2002] Is the expansion of n(n+1)(n+2)...(n+k) known? Page: []

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