5. (10 points) a pythagorean triple is a triple a, b, cof positive integers satisfying the pythagorcan theorem ab2 = 2. a pythagorean triple is called primitive if gcd(a, b, c) = 1 recall from assignment 1 that if a2 +b2 = 2, then a and b cannot both be odd. let a, b, c be a primitive pythagorean triple. then a and b cannot both be even, since otherwisc c is also even and gcd(a, b, c) 2. hence, exactly adopt the convention that a is odd and b is even (and thus c is odd) one of a and b is odd and the other is even. we (a) let s and t be odd integers such that s> t 2 1 and gcd(s, t) = 1. prove that b a= 2 2 forms a primitive pythagorean triple a, b, c (b) let and y be positive integers such that gcd(r, ) 1, and suppose that their product xy is a square. prove that both r and y are squares. (hint. usc the fundamental theorem of arithmetic.) (c) let a, b, c be a primitive pythagorean triple where a, b, c e z, a and c are even. prove that there exist odd integers s and t where s > t> 1and gcd(s, t) = 1 such that a, b,c and show that c+b and c-b are both squares and relatively prime.) (d) find all (not just primitive) pythagoran triples a, b, c of positive integers where odd, and bis 2-b2(cb) (c - b) given by the formulas above. (hint. write a2 are c c 15 =