Meertens number and its variations

TL;DR

This paper explores properties and variations of Meertens numbers, including algorithmic decidability, expanding understanding of these invariant numbers beyond the known example 81312000.

Contribution

It introduces new variations of Meertens numbers and investigates their properties, including the existence decision problem.

Findings

81312000 is the only known Meertens number in base 10

Certain variations allow for finite-time algorithms to determine existence

The paper broadens the understanding of invariance under Godel-like mappings

Abstract

In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Godel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations where there is a finite time algorithm to decide whether such numbers exist.