Proving an equation for a set of ordered triples I am studying for an exam and I would appreciate some help with this question:
Let S be the set of ordered triples of integers that are defined in the following way:


*

*(66, 55, 1331) is an element of S.

*If (a, b, c) is an element of S then (a+7, b+5, 14a-10b+c+24) is an element of S


Prove that for every element (a, b, c) in S:
a^2 - b^2 = c

Hint: try induction.
The first part of an inductive proof of this is almost trivial: simply verify that $66^2-55^2=1331$.
For the second part, you need to assume that $a^2-b^2 = c$, and then prove that $(a+7)^2 - (b+5)^2 = 14a-10b+c+24$. This amounts to messy algebra.
Since every element of $S$ is generated by that operation, this will conclude the proof.
