2 Throughout this question ( \(a , b , c\) ) is a Pythagorean triple with the positive integers \(a , b , c\) ordered such that \(a \leqslant b \leqslant c\).
- Show that \(\mathrm { a } ^ { 2 } = \mathrm { b } + \mathrm { c }\) if and only if \(\mathrm { c } = \mathrm { b } + 1\).
- Create a program to find all the Pythagorean triples ( \(a , b , c\) ) such that \(\mathrm { a } ^ { 2 } = \mathrm { b } + \mathrm { c }\) and \(c \leqslant 1000\). Write out your program in full in the Printed Answer Booklet.
- Write down the number of Pythagorean triples found by your program in (b).
- Prove that there is no Pythagorean triple, \(( a , b , c )\), in which \(\mathrm { b } ^ { 2 } = \mathrm { a } + \mathrm { c }\).