Euler, Stirling, and Wallis: A Case Study in the Notion of Equivalence Between Theorems

The notion that two theorems may be equivalent to each other is sometimes met with hesitation. In this article we tell a story that shows that there is something interesting and useful in this notion. We look at the following three results: Stirling’s formula, Wallis’ product formula, and the evaluation of the probability integral. The task of giving simple proofs of these results is the object of unabated attention. In order to enhance our understanding of these results, we show in a precise way that these results are indeed equivalent to each other.