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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. An Introduction To P-adic Numbers [04/29/2004] What are p-adic numbers? What are they used for? Introducution to Algebraic Numbers and Integers [04/15/2008] While reading about the classifications of complex numbers, such as real, irrational, and so on, I came upon a reference to 'algebraic numbers' and 'algebraic integers'. Can you tell me about those? Inverse Function for Natural Numbers [6/10/1996] I've got a question about the function n = 0.5((a+b)^2+3a+b), which is a one-to-one bijection from pairs (a,b) of natural numbers to single natural numbers n. Irrationality of e+pi and e*pi [09/24/2001] I have read that it is unknown whether either E+Pi or E*Pi is an irrational number. How can we prove that at most one of the two numbers is rational? Irrational Numbers x,y, x^y Rational? [09/28/2001] Are there any irrational numbers x and y such that x^y is rational? The Jumping Frog of Coordinate County [11/03/2009] A frog starts at (0,0) and always jumps a distance of exactly 1 unit to a point with rational coordinates. Show that it is possible for the frog to reach the point (1/5,1/17) and not to reach (0,1/4). Lagrange's Theorem [02/27/2001] In your archives you show proofs of Lagrange's theorem that every positive integer can be expressed as the sum of four squares, but is there an algorithm for identifying which four squares? LaGrange's Theorem [02/24/2001] Please explain LaGrange's Theorem on the number of roots of a polynomial. Large Prime Numbers [12/17/1997] Is there an algorithm to determine whether a very large number is prime? Large Prime Numbers [01/13/2009] What is the largest prime number less than which all primes are known? Largest Mersenne Prime [1/19/1995] What is the largest currently known Mersenne prime? Last Eight Digits of Z [01/18/2002] Consider the recursive function f(n)=13^(f(n-1)), where f(0)=13^13. A=13^13, B=13^(13^13), etc. A=f(0), B=f(A), C=f(B),... Z=f(Y). What are the last 8 digits of Z? Last Four Digits of the Fibonacci Numbers [05/06/2001] Show that there is a number ending with four zeros in the Fibonacci sequence; prove that the Fibonacci sequence has a cycle for the last four digits with a length of 15,000. Linear Congruences of Gaussian Integers [04/11/2003] When does the linear congruence zx congruent to 1 (mod m), for z, x, and m all Gaussian integers, have a solution? Also, when do we say that two Gaussian integers are relatively prime? Linear Diophantine Equation [02/05/2003] Given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5,....., prove that a = b. Linear Proof [11/07/2001] We say that f is linear provided that for every x, y in its domain, f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the set of real numbers), then f is defined by f(x) = cx for some c belong to R. Linear Recurrance Relations [08/10/2004] Is there a general approach to taking a pattern that is defined recursively and finding an explicit definition for it? List of Perfect Numbers [3/24/1995] Do you know any Perfect Numbers besides 6, 28, and 496? Do you know where I can get a list of Perfect Numbers? Mathematical Induction [01/28/2002] Use Mathemetical Induction to prove that any postage of at least 8 cents can be obtained using 3- and 5-cent stamps. Mathematica's PrimitiveRoot[] Function [11/14/2002] My current project is to develop a fast primitive root finder. Maximizing Irregular Polygon Area: Which Circle? [05/01/2011] How do you determine the radius of the circle that maximizes the area of an irregular n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler approaches the question using numerical techniques such as Newton's Method and a binary search, which suggests that no closed-form expression exists. Minimizing the Number of Stamps Needed for Postage [11/20/2007] A student wants to write a computer program to find the minimum number of stamps needed to create a desired postage given the various stamp values that are currently available. Doctor Tom explains why that sort of problem presents a very difficult challenge. Missile Launch Code [08/03/2003] What kind of information could you give all 10 people such that if any 3 of them were to get together, they would be able to launch the missiles, but if there were only 2 of them, the information would be insufficient to figure out the code? Mod, Inverses, and Number Theory [11/27/2001] I already have values for e, p and q, and need to find a value for d in: d = e^-1 mod (p-1)(q-1). Modular Form Ingredients in Fermat's Last Theorem [12/13/2000] The proof of Fermat's Last Theorem shows how the L-series of elliptic curves and the M-series of modular forms correspond, and how modular forms are composed of different "ingredients." Can you provide a description of the ingredients of a modular form? Modular Operators and Multiple Variables? [05/28/2011] A student with six linear equations in two variables wonders if the Euclidean algorithm would solve it. Doctor Vogler simplifies the system by applying the Extended Euclidean Algorithm, and introducing the Chinese Remainder Theorem. Modular Square Roots [03/24/2006] Is there an algorithm for finding modular square roots where the modulus is either a square or an odd composite? Modulus Congruence Proof [04/18/2001] How can I prove 2^(3n+2)+21n = 4 mod (49)? Modulus Proof [04/16/2001] Can you please show me why m^(2^n) = 1 mod(2^(n+2)) when m is an odd integer? More on Order of Operations [02/13/2000] I have found contradictory information on the precedence of the multiplication and division operations. Is there a universally accepted rule for the order of these operations? Multiplication of Integers Modulo (2^16 + 1) [10/18/2002] Prove that 2^16 * 2^15 mod (2^16 + 1) = 2^15 + 1. Multiplicative Groups of Order (p-1) [05/12/2000] What is the proof that primitive roots for multiplicative groups of order (p-1), where p is prime, exist? Is there an algorithm for finding them? No Solution: y^2 = x^3 + 7 [03/17/2003] Show that y^2 = x^3 + 7 has no integer solution. A Number Digits Puzzle [02/23/2001] How can I determine all positive integers with the property that they are one more than the sum of the squares of their digits? Numbers in the Fibonacci Sequence [07/19/2001] How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ? Number Theory and Modular Arithmetic [01/27/2006] If the order of an element mod(p) = n, is it true that the order of this element mod(p^k) = n * p^{k-1}, where p is odd prime and k > 1? Number Theory - Perfect Square [5/26/1996] Find all the possible values of n... Number Theory: Sum of Cube [7/6/1996] Generate the sum of cube for a given integer n... Odd Numbers and Modulo 8 [09/06/2004] Is it true that any odd number greater than 1, multiplied by itself, is congruent to 1 modulo 8? If so, why? The Official Euclidean Algorithm [11/16/2000] Can you state briefly the "official" Euclidean Algorithm? Page: []

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