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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'?

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.

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 O(n) [01/23/2001]
How would you prove that an equation is of order n, or n squared?

Proving Perfect Squares [07/05/1998]
Suppose a, b, and c are positive integers, with no factor in common, where 1/a + 1/b = 1/c. Prove that a+b, a-c, and b-c are all perfect squares.

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

Proving the Associative Property [02/24/2001]
How can I prove that a binary operation is associative, if all I am given is a table for the operation?

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

Proving the Square Root of 2 is Irrational [02/04/2004]
How can you prove that the square root of 2 is irrational using the Rational Root Theorem?

Proving the Square Root of a Prime is Irrational [07/15/1998]
How do you prove that if p is prime, the square root of p is irrational?

Public Key Encryption [03/29/1999]
Examples and discussion of operations used for encryption, including mod.

Pythagorean Quadruplets [12/28/1998]
I am trying to find a formula that generates Pythagorean quadruplets a,b,c,d such that a^2 + b^2 + c^2 = d^2.

Pythagorean Theorem, Fermat's Last Theorem [5/16/1996]
Can the Pythagorean theorem be done with 3 different numbers?

Pythagorean Triple [8/28/1996]
What is the formula for finding the three lengths in a Pythagorean triple where the shortest side is even?

Pythagorean Triples [10/07/1997]
What is a Pythagorean triple?

Pythagorean Triples [04/14/1997]
Why can't all the numbers in a Pythagorean triple be prime?

Pythagorean Triples [07/14/1997]
Is there a formula to determine the solutions to the following equations? a^2 + b^2 = c^2, a^3 + b^3 + c^3 = d^3...

Pythagorean Triples [11/19/1997]
I need to know the first five Pythagorean triples after 3,4,5...

Pythagorean Triples [05/22/1999]
What is the general formula for all sides of any triple?

Pythagorean Triples [05/31/1999]
Is there a procedure for finding Pythagorean triples?

Pythagorean Triples [5/18/1995]
How can the relation between Pythagorean triples be expressed as a formula?

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 Triple with 71 [12/07/1997]
Is there a Pythagorean triple that contains the number 71?

Quadratic Residues [06/30/1998]
I need a fundamental explanation of the concept of quadratic residues.

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...

A Quartic Diophantine Equation: 10657 + 11579x^2 + x^4 = y^2 [12/29/2008]
Doctor Vogler helps a student look for integer solutions to a quartic polynomial by noticing a difference of squares in its coefficients and factoring its constant term.

Ramsey's Theorem and Infinite Sequence [06/01/1999]
Ramsey's Theorem applied to divisibility in infinite sequences.

Rational and Irrational Numbers: Multiplication, Division [10/15/2001]
I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational.

Rationalizing a Denominator with Multiple Cube Roots [04/22/2011]
A student of field theory wonders how to remove the cube roots from the denominator of 1/(a + b*CBRT(q) + c*CBRT(q)^2). Building on the conjugacy of square roots, Doctor Vogler writes out the required conjugates.

Real and Rational Numbers [02/27/2001]
How can I show that the number of rational numbers between 0 and 1 is the same as the number of natural numbers (considering the ordering of fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?

Real Numbers [08/08/1997]
What exactly is a real number?

Reasoning about Integers [10/06/2004]
When positive integers p and q are divided by an even positive integer t, they have remainders 2 and t/2, respectively. What is the remainder when the product pq is divided by t?

Reciprocals of Integers Greater Than 1 as Sum of a Series [07/01/2004]
Show that the reciprocal of every integer greater than 1 is the sum of a finite number of consecutive terms of the series 1/[j(j + 1)].

Rectangular Solids from Blocks [09/25/1998]
How many rectangular solids can be made from "n" cube-shaped blocks?

Recurrence Relation for a Pell Equation [11/09/1999]
Can you help me find a recurrence relation for generating solutions to the Pell equation x^2 - 5y^2 = 1?

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 [10/07/1999]
What does the term relatively prime mean, and how can you determine if two numbers are relative primes?

Relatively Prime Pythagorean Triples [09/13/1997]
Questions about Pythagorean triples.

A Remainder Riddle with Relatively Prime Divisors [06/18/2016]
A teen wonders what smallest positive integer satisfies three related divisibility criteria. Doctor Greenie addresses all the required remainders simultaneously in a first approach; then outlines a piecemeal method.

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.

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