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.

翻译：忙碌海狸数的理论存在性为可判定性提供了一个新概念，并为猜想提供了相应的启发式方法。一个猜想能被建模所需的最少状态数，给出了描述该猜想的逻辑系统的分类标准。在本工作中，我们构造了显式图灵机，用于搜索大于7的布罗卡问题解以及第4个费马素数之后的素数，这些图灵机当且仅当存在这样的解时才会停机。