Fermat’s Last Theorem and Related Problems

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 Furtw¨angler theorems are reformulated. Hasse used one of his reciprocity laws to give a more systematic proof of Furtw¨angler’s theorems. Hasse’s reciprocity law is modified to deal with a certain condition. Vandiver’s theorem is reformulated and generalized. The eigenvalues of 2p x 2p matrices for the p = 3 case are investigated. (There is a relationship between the modularity theorem and a re-interpretation of the quadratic reciprocity theorem as a system of eigenvalues on a finite-dimensional complex vector space.) A generalization involving generators and \reciprocity" has solutions for every p value.

Using decomposition groups to prove theorems about quadratic residues

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences

Consider the system x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X2 + Y7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively.

4. Congruences, clocks, and calendars

‘Congruences, clocks, and calendars’ demonstrates how we might apply the idea of congruence, first introduced by Gauss in 1801, to problems such as testing which Mersenne numbers are primes and finding the day of the week on which a given date falls. Ancient Chinese puzzles depended on the solving of simultaneous linear congruences, inspiring mathematicians and giving rise to the Chinese Remainder Theorem. Exploring quadratic congruences leads towards the law of quadratic reciprocity, noted by Euler and Legendre and proved by Gauss. The problem, ‘Is 1066 a square or a non-square?’ can be solved by applying this law several times to reduce the numbers involved.