A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES

Let a,b,cbe a primitive Pythagorean triple and set a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}, where mand nare positive integers with m>n, \text{gcd}(m,n)=1and m\not \equiv n~(\text{mod}~2). In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation (m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}is (x,y,z)=(2,2,2). We use biquadratic character theory to investigate the case with...