On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples

TL;DR

This paper investigates divisibility restrictions in Fermat's Last Theorem, focusing on how these restrictions influence the existence of solutions across various number systems and emphasizing the role of the variable c in solution exclusion.

Contribution

It introduces a novel analysis of divisibility and coprimality conditions, highlighting the impact of the variable c and extending considerations to complex and quaternion solutions.

Findings

Divisibility restrictions reduce the set of Fermat solutions.

Exclusion of solutions is linked to gcd properties and divisibility conditions.

Odd powers over quaternions do not satisfy Fermat-like equations.

Abstract

The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat triples. In our analysis, the restriction on positive integer (PI) solutions ((a,b,c,n) up to the point when there is no more) is not along with restriction on power n in PI as decreasing sets {PI } containing {odd} containing {primes} containing {regular primes}, etc. as in the literature, but with respect to exclusion of more and more c in PI as increasing sets {primes p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation is analyzed in relation to exclusion of solutions, and the effect of simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on the decrease of cardinality of solutions is exhibited. Again, our derivation focuses mainly on the variable c rather than on variable n, oppositely to the literature in which the FLT is historically separated via the values of power n. Among the most famous are the known, about 2500 years old, existing Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases (of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat) less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion (a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over quaternions breaks.