Measuring Decidability as Related to Busy Beaver Numbers

TL;DR

This paper introduces a novel approach to decidability using Busy Beaver numbers, constructing Turing machines that verify specific number-theoretic conjectures by halting only if solutions exist.

Contribution

It presents explicit Turing machines linked to Busy Beaver numbers that test conjectures like Brocard's problem and Fermat primes, connecting computational limits to logical classification.

Findings

Constructed Turing machines for Brocard's problem and Fermat primes.

Linked Busy Beaver numbers to the classification of logical systems.

Provided a heuristic for conjecture decidability based on state complexity.

Abstract

The theoretical existence of Busy Beaver numbers provides a new notion for decidability and corresponding heuristic for conjectures. The minimum number of states in which a conjecture can be modeled gives a classification of what logic system can describe said conjecture. In this work, we construct explicit Turing machines that search for a solution to Brocard's problem greater than 7 and a Fermat prime beyond the 4th which halt if and only if such a solution exists.