This puzzle was MEI’s Maths Item of the Month in August 2011. An archive of puzzles going back to 2006 is available on the MEI website. You can also view the puzzles categorised by GCSE and A level Mathematics topics. The puzzle below is aimed at AS level students, but would also be accessible to bright Year 11 students.
maths equation is a Pythagorean triple.
Add the reciprocals of any two consecutive odd numbers. Will the resulting fraction, maths, always generate an integer Pythagorean triple, maths?
By writing the consecutive odd numbers as maths and maths the sum of the fractions is: maths equation
The conjecture is that this generates a fraction maths such that maths where a, b and c are all integers.
Squaring and adding the terms for a and b gives: mahts equation
maths equation will be an integer, therefore maths equation is a Pythagorean triple.