The question is:- does 12 divide the product of the pythagorean triangle?
My answer is that yes,
since
I don't know if this is correct, but how can I prove this.
2)
if and , show that a and b are nth powers
The question is:- does 12 divide the product of the pythagorean triangle?
My answer is that yes,
since
I don't know if this is correct, but how can I prove this.
2)
if and , show that a and b are nth powers
This is for the case
If r, s, and t are positive integers such that (r,s) = 1 and , then there are integers m and n such that and .
If r = 1 or s = 1, then this is obviously true, so we may suppose that r > 1 and s > 1. Let the prime-power factorizations of r, s, and t be
and
.
Because (r,s) = 1, the primes occurring in the factorizations of r and s are distinct. Because , we have
From the fundamental theorem of arithmetic, the prime-powers occurring on the two sides of the above equation are the same. Hence, each must be equal to for some j with matching exponents, so that . Consequently, every exponent is even, and therefore is an integer. We see that and , where m and n are the integers
Maybe this will help make a general case.