Studying solutions to Diophantine equations using the table of $N$th digital roots of integers

TL;DR

This paper explores using digital roots to analyze Diophantine equations, providing a table of digital root operations to help determine solvability, with a minor extension of Fermat's little theorem.

Contribution

It introduces a digital root-based method for assessing the solvability of Diophantine equations, offering a practical tool despite its limitations.

Findings

Digital root tables assist in solvability checks

Method helps identify non-solvable equations

Minor extension of Fermat's little theorem proposed

Abstract

In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of obtaining complete solutions of equations, but it is frequently useful in determining whether this equation is solvable in integers or not. A minor extension of Fermat's little theorem is put forward.