# Report on fortnightly maths prize question for 20 January 2015

You know 3,4,5 is a “Pythagorean triple”: the numbers could be the sides of a right-angled triangle. You probably know that 5,12,13 is a Pythagorean triple, too. Your task: to find out as much as you can about what other sets of three whole numbers are Pythagorean triples.

The prize was won by Hamse Adam.

A bit of arithmetic shows you that there is a sequence of Pythagorean triples like:
3,4,5
5,12,13
7,24,25
9,40,41
11,60,61
….
The first number goes up by 2 each time. The second number are four times the triangle numbers, and go up by 4n (from the [n−1]’th to the n’th triple]. The third number is the second number plus 1.

The general formula is 2n+1,2n(n+1),2n2+2n+1

These are all “primitive” Pythagorean triples, meaning that the whole numbers have no common factors.

From any “primitive” Pythagorean triple, you can get lots of others. For example, 3,4,5 gives 6,8,10 and 9,12,15 and 12,16,20… which are all Pythagorean triples.

The sequence above does not contain all the “primitive” Pythagorean triples. For example, there is 8,15,17.

Every primitive Pythagorean triple has an odd hypotenuse, and the other sides are one odd, one even. The even side is always not just even, but divisible by 4.

If you multiply together the three numbers of any Pythagorean triple, the answer is always divisible by 60.

a, b, c is a primitive Pythagorean triple if a, b, c are whole numbers, they have no common factors, and:

c2 = a2 + b2

Examples: 3, 4, 5; 5, 12, 13; 8, 15, 17

All the Pythagorean triples can be got as ka, kb, kc, where k is a whole number and a, b, c is a primitive Pythagorean triple (PPT).

Example: 6, 8, 10 = 2×3, 2×4, 2×5.

We can find a rule to generate all the PPTs. Continue reading