Finding Permutiples of a Known Base and Multiplier

TL;DR

This paper introduces two graph-theoretical methods to find permutiples of a given base and multiplier without needing prior examples or digit knowledge, advancing the search for such numbers.

Contribution

It presents novel graph-based techniques for discovering permutiples directly from base and multiplier, eliminating the need for known examples.

Findings

Developed two new methods for finding permutiples

Methods do not require prior digit examples

Enhanced the ability to identify permutiples in various bases

Abstract

Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of their digits (palintiples). Previous efforts have produced methods which construct new examples of permutiples with the same set of digits as a known example. Using simple graph-theoretical and finite-state machine constructions, we advance previous work by describing two methods for finding permutiples of a known base and multiplier with no need for known examples or prior knowledge of digits.