History of Numbers

Pythagorean triples

Pythagorean triples are sets of numbers which obey Pythagoras' Theorem:

For instance, the numbers 3, 4 and 5 are a Pythagorean triple, so are the numbers 5, 12 and 13.

• Find as many Pythagorean triples as you can.
• Can you find a way to generate them?

Pythagorean triples and Fibonacci numbers

Fibonacci numbers are formed by adding the two preceding numbers in the sequence. If you start with 1 and 1, the sequence you generate is:

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

You don't have to start with 1 and 1, however. You can start with any numbers you like.

Look at any four consecutive Fibonacci numbers, F_n, F_n+1, F_n+2, and F_n+3.

• form the product of F_n and F_n+3 - put this equal to p, say
• form the product of F_n+1 and F_n+2, and double it - put this equal to q, say
• now square both p and q and add them
• what do you notice?
• investigate this for other sets of four consecutive Fibonacci numbers

Problems involving Pythagorean triples or Fibonacci numbers

There are many problems on the Nrich website involving Pythagorean triples and Fibonacci numbers. Use the link to go to the Nrich website, then put Pythagorean triples or Fibonacci numbers into the Search engine - Third Tier problems are likely to be about the right level. Have a go at some of the problems.