In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl gives a good general criterion for independence from that fragment in ``Diophantine equations in a fragment of number theory'' in the book Computation and Proof Theory, Springer Lecture Notes in Mathematics Volume 1104, 1984, pp 389-398.

Is FLT currently known to be independent of any larger fragment of PA?