​Fermat's Last Theorem

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Fermat's last theorem states that the equation x^n+y^n=z^n admits no positive integral solutions (x,y,z). Even though the theorem was proven by A. Wiles in 1993 using hard modern techniques from the field of elliptic curves (over 355 after first stated by Fermat), the question of the existence of an elementary proof still excites mathematicians untill today (Fermat mentioned that such a proof exists - which might add to the interest).. 

In a recent study  (
On Fermat curves modulo a finite number  https://arxiv.org/abs/1707.01856 ) we investegated "mock solutions" of Fermat's equation - which are pairs of positive integral numbers (x,y) such that x^n+y^n can be divided by z^n, or more generally z^j, for any j>0 (for a fixed positive integer z - to which we refer as the radius). We refer to the resulting subset of Z^2 - as the j-th Fermat tile of radius z - denoted by T_n(z ; j). Note that a geniune solution of Fermat's equation is, of course, a "mock solution - but not vice versa.

Remarkably - Fermat tiles are actually highly structured, and this structure can be neatly described in terms of elementary number theory (actually, results which are due to Fermat himself). For example - the following is an illustration T_4(17 ;2) the second Fermet tile of radius 17: 

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The different colors in the picture - come to represent the fact that the non-linear Fermat tiles T_n(z;j) actually split into n linear-tiles (observed in the manuscript and proved in Proposition 2.6). A study of the algebraic properties of the tiles leads to heavy new restrictions on the "existence" of genuine solutions of Fermat's equation - specifically - such a solution would have to satisfy a robust system of double recursion relations - to which we refer as the "zipper" relations (Theorem B in the mauscript). A video presentation is available below - the ​files can be viewed here​  Fermat-Presentation.pdf