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Browse College Number Theory
Stars indicate particularly interesting answers or
good places to begin browsing.
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^2p)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 nonzero 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 abs2^(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 + (p1) 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 nonresidue 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^((p1)/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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