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Let $a,b,c$ be relatively prime positive integers such that$a^{2}+b^{2}=c^{2}.$ In 1956, Je\'{s}manowicz conjectured that for any positiveinteger $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positiveintegers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$be a Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture istrue for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.

Je\'{s}manowicz' conjecture and Fermat numbers