## Math: Pythagorean triples and Fermat’s Last Theorem

Tutoring math, ideas are always important. The tutor mentions Fermat’s Last Theorem. A Pythagorean triple is a set of three integers that satisfy a2+b2=c2 There are infinite Pythagorean triples. However, there are no corresponding sets of integers for higher n: …

## Number theory: Fermat’s Theorem

The tutor introduces Fermat’s Theorem with a first example. Fermat’s Theorem states that, for a prime number p and a number b not a multiple of p, bp-1 ≡ 1 (mod p). (See my post here for a working definition …

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## Number theory: congruence: another problem from Underwood Dudley

The tutor investigates a problem involving the remainder of a power. On page 48 of his Elementary Number Theory, second edition, Underwood Dudley requests the remainder when 20012001 is divided by 26. Solution: 2001 mod 26 = 25 ⇒ 2001 …

## Number theory: another problem from Dudley’s Elementary Number Theory

The tutor investigates a problem involving composite numbers. For problem 4b, page 19, of his Elementary Number Theory (second edition), Dudley invites the reader to prove there are infinite n such that both 6n-1 and 6n+1 are composite. (Composite means …

## Math: linear congruences

The tutor solves a system of linear congruences. Back in my post from March 25, 2014, I explain that “mod” means remainder: for instance, 7 mod 3 = 1. Two numbers that, divided by a number n, give the same …

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## Math: number theory: (n mod 3)²

The tutor shows an interesting consequence of mod 3 arithmetic. Back in my March 25, 2014 post, I mentioned that mod means remainder. For example, 19 mod 4 = 3, because when you divide 19 by 4, you get 3 …

## Math: number theory: linear combinations that sum to 1

The tutor tackles an age-old proof in a new (to him, anyway) manner. A famous theorem of number theory goes like this: For the integers a and b, there exists a solution with integers x and y to ax+by=1 if …

## Math: Pythagorean triples: proof of yesterday’s generating formulas

The tutor shows that yesterday’s formulas to generate Pythagorean triples are valid. In yesterday’s post I showed a way to generate Pythagorean triples x, y, z from an odd number n: x n y (n²-1)/2 z (n²+1)/2 Let’s make sure …

## Math: number theory: a formula for generating Pythagorean triples

The tutor continues his discussion about Pythagorean triples. Back in my January 7, 2016 post I brought up Pythagorean triples, which are all-integer solutions to x² + y² = z² The equation above is based on the familiar a² + …