A related conjecture from Euler
Noam Elkies gave a counterexample, namely 
. Subsequently, Roger Frye found the
absolutely smallest solution by (more or less) brute force: it is
. "Several years",
Math. Comp. 51 (1988) 825-835.
This synopsis is quite brief. A full survey would run too many pages.
References
[1] J.P.Butler, R.E.Crandall,& R.W.Sompolski, Irregular Primes to One Million. Math. Comp., 59 (October 1992) pp. 717-722.
Fermat's Last Theorem, A Genetic Introduction to Algebraic Number Theory. H.M. Edwards. Springer Verlag, New York, 1977.
Thirteen Lectures on Fermat's Last Theorem. P. Ribenboim. Springer Verlag, New York, 1979.
Number Theory Related to Fermat's Last Theorem. Neal Koblitz, editor. Birkhäuser Boston, Inc., 1982, ISBN 3-7643-3104-6