TITLE:
Small Modular Solutions to Fermat’s Last Theorem
AUTHORS:
Thomas Beatty
KEYWORDS:
Fermat’s Last Theorem, Modular Arithmetic, Congruences, Prime Numbers, Primitive Roots, Indices, Ramsey Theory, Schur’s Lemma in Ramsey Theory
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.10,
October
31,
2024
ABSTRACT: The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers
x,y,z
to the equation
x
n
+
y
n
=
z
n
for
n>2
. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime
p
0
such that for all primes
p≥
p
0
the congruence
x
n
+
y
n
≡
z
n
(
modp
)
has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.