Let
$a, b, c$
be relatively prime positive integers such that
${a}^{2} + {b}^{2} = {c}^{2} $
. In 1956, Jeśmanowicz conjectured that for any positive integer
$n$
, the only solution of
$\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $
in positive integers is
$(x, y, z)= (2, 2, 2)$
. In this paper, we consider Jeśmanowicz’ conjecture for Pythagorean triples
$(a, b, c)$
if
$a= c- 2$
and
$c$
is a Fermat prime. For example, we show that Jeśmanowicz’ conjecture is true for
$(a, b, c)= (3, 4, 5)$
,
$(15, 8, 17)$
,
$(255, 32, 257)$
,
$(65535, 512, 65537)$
.