I have some Pythagorean Triples proofs that I kind of have a hard time getting started on. I am just looking for a little insight to help me get started. Any help at all would be greatly appreciated. Here goes.
A pythagorean triple is a triple of positive integers (a,b,c) such that a^2 +b^2=c^2. In othere words, a set of three numbers is a pythagorean tiple if there exists a right triangle having these three thtegers as its side lengths. The triples one most commonly encounters when one first studies right triangles are (3,4,5) and (5,12,13).
(a) Are there othere pythagorean triples besides (3,4,5) which consist of three consecutive positive integers? Either give an exam;le of another such triple, or else prove n o other such triple can exist.
(b) Prove that there are infinitely many pythagorean triples (a,b,c) for which c -b=1. (Suggestion: direct proof by constructing them explicitly. Start by finding some small ones and look for a pattern. )
(c) Prove that there does not exist a pythagorean triple (a,b,c) such that a is odd, b is odd, and c sis even. ( Suggestion: Assume (a,b,c) is such a triple and derive a contradiction.)
On (a) it is pretty obvious that there is not any more triples that consist of three consecutive positive integers as a^2 + b^2 gets bigger much quicker than c^2. I think that this is proof by contradiction but not completely sure. Like I said any help would be greatly apprecitated.