Fermat’s Last Theorem and Related Problems
Darell Cox *
Department of Mathematics, Grayson County College, United States.
Sourangshu Ghosh *
Department of Civil Engineering, Indian Institute of Technology Kharagpur, India.
Eldar Sultanow *
Potsdam University, Chair of Business Informatics, Processes and Systems, Potsdam, Germany.
*Author to whom correspondence should be addressed.
Abstract
Empirical evidence in support of generalizations of Fermat's equation is presented. The empirical evidence consists mainly of results for the p = 3 case where Fermat's Last Theorem is almost false. The empirical evidence also consists of results for general p values. The \pth power with respect to" concept (involving congruences) is introduced and used to derive these generalizations. The classical Furtwangler theorems are reformulated. Hasse used one of his reciprocity laws to give a more systematic proof of Furtwangler's theorems.
Keywords: Fermat’s last theorem, modularity, quadratic reciprocity, Furtw¨angler theorems, Hasse’s reciprocity law