Let be positive integers. Let be a primitive Pythagorean triplet with . In
1956, L. Jesmanowicz conjectured that the equation
has only the positive integer solution
. In this paper, using certain elementary methods,
we prove that if
, then
the above equation has only the positive integer solution
. Thus it can be seen that Jesmanowicz's conjecture
is true for infinitely many primitive Pythagorean triplets.