Jeśmanowicz' conjecture on exponential diophantine equations

Let (a,b,c)be a primitive Pythagorean triple such that a^2+b^2=c^2with even b In 1956, L. Jeśmanowicz conjectured that the equation a^x+b^y=c^zhas only the solution (x,y,z)=(2,2,2)in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution (x,y,z)with even x,zthen x/2and z/2are...