A determinant connected with Fermat’s last theorem

Certain Congruence Criteria Connected with Fermat's Last Theorem.

Fermat’s Last Theorem

Fermat learned his number theory from the books of Diophantus; it was in the margins of his copy that he wrote down that he had discovered a truly marvelous proof of the fact that X + Y n = Z has no solutions in natural numbers for n > 2, and that the margin was too small to contain it. The books of Diophantus, including these remarks, were published posthumously by his son. By the early 1800s,...

Fermat's Last Theorem

The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J....

The Last Player Theorem

Game trees are an important model of decision-making situations, both in artificial intelligence and decision analysis, but many of the properties of game trees are not well understood. One of these properties is known as biasing: when a minimax search is done to an odd search depth, all moves tend to look good, and when it is done to an even search depth, all modes tend to look bad. One explan...

A noncommutative generalization of Auslander's last theorem

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...