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Browse College Number Theory
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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 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.

Relationship Between GCF and LCM [05/22/2002]

What is the exact relationship between the gcf or gcd and the lcm of two numbers?

Relatively Prime Pythagorean Triples [09/13/1997]

Questions about Pythagorean triples.

Remainders, Pigeons, and Pigeonholes [03/26/2003]

Given 17 integers, prove that it is always possible to select 5 of the 17 whose sum is divisible by 5.

Remainder when Dividing Large Numbers [04/17/2001]

How can I find the remainder when (12371^56 + 34)^28 is divided by 111?

Repeating Decimals [05/14/1997]

If the length of the repeating sequence in a decimal of a converted fraction is less than the denominator of the fraction, is it always an integer factor of the denominator minus one?

Repeating Digits of Fractions [04/28/1999]

Do you know any theorems relating to the length of the repeating portion of the decimal representation of fractions?

Representing Positive Integers in an Irrational Base [08/13/2007]

I know that 3 in base 2 is written as 11. But how would I express 3 in terms of an irrational base, like base square root of 2?

Residues and Non-Residues [05/04/2003]

If p > 3, show that p divides the sum of its quadratic residue.

Reverse Modulus Operator [10/09/2001]

Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

RSA Encryption [04/25/2002]

Decrypt the encrypted message in ciphertext C to find the original plaintext, a string of English letters.

Second-Order Linear Recurrences [06/08/2001]

Three problems involving recurrence equations.

Second Order Recurrence with Non-Constant Coefficients [05/27/2005]

I'm trying to find a closed form solution of a second order recurrence relation with no constant coefficients, specifically: u(n+2) = 2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?

Shanks-Tonelli Algorithm [01/17/2001]

How can I calculate the four square roots of a number modulo n, where n is the product of two primes p and q?

Showing Two Numbers Are Relatively Prime [08/01/2008]

Show that for every natural number n, 21n + 4 and 14n + 3 are relatively prime.

Simultaneous Modulus Congruencies [04/18/2001]

How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?

Slot-wise Addition of Pythagorean Triples [07/17/2003]

Is it possible to have a primitive Pythagorean triple, (a,b,c) such that a^2+b^2 = c^2, and two other Pythagorean triples, not necessarily primitive (x,y,z) and (p,q,r) with the property that a=x+p, b=y+q, and c=z+r?

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