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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'? Product of Two Primes [10/27/1999] How many positive integers less than 100 can be written as the product of the first power of two different primes? Products of Integers (Even or Odd) [03/23/2002] How can I prove that the product of two even integers is an even integer and the product of two odd integers is an odd integer? Programs to Find Prime Numbers [11/27/1996] Can a program be written in BASIC to compute the number of prime numbers smaller than n? Program to Convert Number Bases [07/12/1999] Is there an easier method for converting bases than dividing and collecting the remainders? I want to write a computer program to do this. Proof by Contraposition [03/06/2002] How can I prove that n^6 + 2n^5 - n^2 - 2n is divisible by 120? Proof by Induction [05/24/2002] Prove by induction that (n^7 - n) is divisible by 42. Proof by Mathematical Induction [09/24/1999] Prove the following statement by mathematical induction: for any integer n greater than or equal to 1, x^n - y^n is divisible by x-y where x and y are any integers with x not equal to y. Proof Involving Legendre Symbol [02/03/2003] If p, q are both prime odd numbers such that they are not factors of a, and p=q(mod 4a), prove that (a/p)=(a/q). Proof Involving mod 5 [10/27/2002] Prove n^2 mod 5 = 1 or 4 when n is an integer not divisible by 5. Proof of Lagrange's Theorem [11/23/2000] I am looking for a proof of Lagrange's Theorem, which states that any positive integer can be expressed as the sum of 4 square numbers. Proof of the Infinite Series That Calculates 'e' [02/04/2004] Is there a proof about this infinite series that gives the value of e: 1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n! where n goes to infinity? Proof of the Rational Root Theorem [11/13/2000] How can I prove the Rational Root theorem? Proof Regarding LCM [12/05/2001] Is there a proof of the equation: given integers a and b, a*b = GCF(a,b) * LCM(a,b)? Proofs that Every Natural Number Factors into Only One Unique Product of Primes [03/15/2012] How do we know that every natural number has one, unique factorization? After identifying this as the Fundamental Theorem of Arithmetic and mentioning a proof by Euclid, Doctor Jacques sets up a simpler proof that relies on contradiction. Proof That 0/0 = 1 Based on x^0 Equaling 1? [02/02/2006] I know the reason x^0 = 1 is because 1 = (x^3)/(x^3) = x^(3-3) = x^0. I also know that 0/0 doesn't make sense, but following a similar argument for 0^0 gives 1 = (0^3)/(0^3) = 0^(3-3) = 0^0. But since (0^3) = 0, haven't I just shown that 0/0 = 1? Proof That 2 Does Equal 1! [03/24/1997] I came up with a proof that 1 = 2. Where does my math go wrong? Proof that an Even Number Squared is Even [06/02/1999] How do you prove that any even number squared is even and any odd number squared is odd? Proof That Equation Has No Integer Roots [05/09/2000] How can I prove that if p is a prime number, then the equation x^5 - px^4 + (p^2-p)x^3 + px^2 - (p^3+p^2)x - p^2 = 0 has no integer roots? Proof That Product is Irrational [03/28/2001] How can I prove that the product of a non-zero rational number and an irrational number is irrational without using specific examples? Proof That sin(5) is Irrational [04/24/2001] How do you prove that sin(5) is an irrational number? Proof that Sqrt(3) is Irrational [08/14/1997] How does one prove that sqrt(3) is irrational? or others? Is there a general algorithm? How about just for primes? Proof That the Cube Root of 3 is Irrational [05/22/2000] How can I show that the cube root of 3 is irrational? Proof with Pigeonhole Principle [09/20/2001] Prove that among five points selected inside an equilateral triangle with sides of length 2, there always exists a pair at a distance not greater than 1. Proof with Powers of 2 and a Product [10/22/2008] Prove that starting with any power of 2 there is a number such that the product of those two numbers will contain only the digits 1 and 2. For example, 4*3 = 12, 8 * 14 = 112, 64*33 = 2112. Properties of the Phi Function [01/19/1999] What are some properties of the phi function? What about the phi function and prime numbers? Prove 101 the Only Prime [12/14/2002] In this sequence of integers, all in base 10: 101, 10101, 1010101, 101010101, 10101010101, .......,,,, etc., prove that 101 is the only prime in the sequence. Prove a and b are Perfect Squares [12/28/2001] Let a and b be positive integers such that (a,b) = 1 and ab is a perfect square. Prove that a and b are perfect squares. Prove a = b = c [01/27/2002] When a^2 + b^2 + c^2 = ab + bc + ca and abc does not equal 0, prove that a = b = c. Prove that 7 + 17*sqrt(17) is Irrational [09/22/2002] I need to prove that 7 + 17*sqrt(17) is irrational. I know to set it equal to a/b, and have already proven that any square root of a prime number is irrational. Prove That an Expression is a Multiple of 10 [12/19/2002] If a and b are positive integers, prove that (a^5)*(b) - (a)*(b^5) is a multiple of 10. Prove that Log A is Irrational [06/14/1998] Can you help me prove that a common log of a number (not powers of 10) is irrational? Prove that n^3 + 2n is Divisible by 3 [04/15/2001] Prove by mathematical induction that n^3 + 2n is divisible by 3 for all positive integers. Prove Twin Primes Greater Than 3 Divisible by 12 [10/08/2002] Prove that if p and q are twin primes, each greater than 3, then p+q is divisible by 12. Prove x^2+y^2 Not Divisible by 4 [09/20/2001] Prove that if x and y are odd, then x^2 + y^2 is even but not divisible by 4. Proving a Number is Prime [10/13/2004] How do you really prove that 2 or some other number is a prime number? Proving a^x = a^y iff x = y [12/13/2000] How can I prove that a^x = a^y iff y = x for all real numbers x and y? Proving De Moivre's Theorem [12/03/1997] Prove De Moivres theorem: - (cos(x)+isin(x))^n = cos(nx) + isin(nx) . Proving Divisibility [04/30/2002] Prove that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 15. Proving Divisibility [09/11/2003] Prove that (n^2 - n) is divisible by 2 for every integer n; that (n^3 - n) is divisible by 6; and that n^5 - n is divisible by 30. Proving e is Irrational [11/19/1997] My professor suggested using a proof by contradiction, but I don't understand how to do it. Page: [<]

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