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Browse College Number Theory
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Selected answers to common questions:
    Testing for primality.

Proof: Infinitely Many Primes [6/24/1996]
I need a proof that there are an infinite number of primes other than Euclid's proof by contradiction. What is the Dirichlet proof?

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 Irreducibility in Z_p[x] [11/03/2006]
Let f be a primitive polynomial in Z_[x]. For any prime p, let f_p be the image of f under the map Z[x]--->Z_p[x]. Then f is irreducible iff f_p is irreducible in Z_p[x] for some prime p.

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 No Integer Solutions for a^3 + b^3 = c^3 [02/08/2004]
In the equation a^3 + b^3 = c^3, how is it possible to prove that there are no integers that satisfy the equation?

Proof of the Compositeness Theorem [04/10/2005]
I have determined that there are no prime numbers in the interval [n! + 2, n! + n], but I am trying to make sense of why. Is there a theorem that expalins why this works?

Proof that 1 + 1 = 2 [06/10/1999]
Can you prove that 1 + 1 = 2?

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 the Cube Root of 3 is Irrational [05/22/2000]
How can I show that the cube root of 3 is irrational?

Proof with Exponential Diophantine Inequality [03/10/2008]
Claim: Let m and n be positive integers. Then abs|2^(n+1/2) - 3^m|<1 if and only if n = m = 1. I would like to know how to prove that claim or, at least, obtain some hints as to how to proceed.

Prove Mersenne Number is Prime or Pseudoprime [10/11/2008]
Let p be a prime number. Prove that 2^p - 1 is either a prime number or a pseudoprime number (2^n is congruent to 2 modulo n, where n is composite).

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 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.

Proving a Polynomial is a Perfect Square [08/17/2007]
Let a and b be odd integers such that a^2 - b^2 + 1 divides b^2 - 1. Prove that a^2 - b^2 + 1 is a perfect square.

Proving a Polynomial is Irreducible using Eisenstein's Criterion [10/18/2004]
Let p be a prime number. Show that the polynomial x^p + px + (p-1) is irreducible over Q if and only if p >= 3.

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 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 Fermat's Last Theorem for N = 4 [05/18/2000]
How can you prove Fermat's Last Theorem for the specific case n = 4?

Proving Infinite Primes [05/01/2008]
What is a proof that there are infinitely many primes of the form 4n + 1?

Proving Phi(m) Is Even [04/22/1998]
Explain why phi(m) is always even for m greater than 2...

Proving That Z_{mn} is Isomorphic to Z_m X Z_n [04/22/2009]
If m and n are relatively prime, show that Zmn is isomorphic to Zm X Zn.

Proving the Convergence of Continued Fractions [01/10/2001]
How do you prove that the sequence of convergents 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/...)))) actually converges to pi?

Proving Theorems [07/20/2001]
False statements of Euler's Theorem and Fermat's Little Theorem.

Proving the Properties of Natural Numbers [03/08/2000]
How can you prove or derive the commutative, associative, and distributive properties of numbers?

Public Key Cryptography [06/18/1997]
Resources for learning about public key cryptography (RSA system).

Pythagorean Triples [8/25/1996]
I am looking for a triple of 3 natural numbers (a,b,c)...

Pythagorean Triples [07/10/1997]
Why are (3,4,5), (20,21,29), (119,120,169), and (696,697,985) considered Pythagorean triples?

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 Triples (x,c,y) with Fixed c [05/09/2003]
Is there a shortcut to finding the integer solutions to equations of the type x^2 + c = y^2, where c is a constant of known value?

Quadratic Diophantine Equation [01/16/2009]
Find all positive integers N such that 2*N^2 - 2*N + 1 is the square of an odd integer.

Quadratic Number Fields and Integer Solutions [08/16/2007]
Prove that the equation 34*y^2 - x^2 = 1 in Z (integer number set) has no solution.

Quadratic Polynomial Number Theory and Number Fields [05/12/2004]
I have the polynomial P(x) = 2*x^2 + 3*x + 4, and I'm trying to find all values of x for which P is a perfect square. Are there infinite values of x that generate perfect squares for P? Is there a formula to generate those x values? From there, is there a general formula for P(x) = a*x^2 + b*x + c?

Quadratic Residue Equations [11/21/2009]
Devise a method for solving the congruence x^2 == a (mod p) if the prime p == 1 (mod 8) and one quadratic non-residue of p is known.

Quadratic Residues [1/12/1995]
Are quadratic residues used only to prove other algorithms, or is there actually a useful application in solving, for example, numerical problems?

Quadratic Residues [05/24/2002]
If p is prime, and if a^((p-1)/2) is congruent to 1 modulo p, then show that a is a quadratic modulo p.

Quadratic Residues [03/19/2004]
Given x^2==a(mod p), let p be an odd prime. There are exactly (p - 1)/2 incongruent quadratic residues of p and exactly (p - 1)/2 quadratic nonresidues of p. Can you provide an example that helps explain this concept?

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...

Rational Solutions to Two Variable Quadratic Equation [11/25/2003]
Find all the rational solutions to x^2 + y^2 = 2.

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