Pyramids and Squares

I have been spending my free time the last few days on the task of working backwards through three proofs in a 19th century French language mathematics journal. This started with a simple question in the G+ Mathematics community, posted by Jeremy Williams: “Who can find the largest tetrahedral number that is also a square?”

The basic proof for this is from “Question 1194” by A. J. J. Meyl, Nouvelles Annales de Mathématiques, S. 2, V. 17 (1878), 464-467. The framing of the answer is more casual than Williams’s mathematically rigorous version: “A pile of balls with a triangular base does not contain a number of balls that is square unless it has a base side of 1, 2, or 48.”

Meyl’s proof relies on a proof from the preceding volume of the Annales, Édouard Lucas’s “Question 1180” (429-432), in which he concludes: “A pile of balls with a square base does not contain a number of balls that is square unless it has a base side of 24.”

Lucas’s proof, in turn, relies on a proof from the same volume of the Annales, MM. Gerono’s “Question 1177” (230-234), which finds all solutions for the equation \(y^2 = x^3 + x^2 + x + 1\). These solutions are \((x, |y|) = \{(-1, 0), (0, 1), (1, 2), (7, 20)\}\).

Working through these three proofs has been an interesting review of how mathematical ideas build on each other. The connection between Meyl’s question (can a pile of balls with a triangular base have a square number?) and Lucas’s question (can a pile of balls with a square base have a square number?) is obvious, even if the answer isn’t immediately so. However, the connection between these two and Gerono’s question (what are the solutions for  \(y^2 = x^3 + x^2 + x + 1\)?) may not be so immediate.

Also interesting is which solutions Meyl and Lucas include and ignore. Lucas includes both of the trivial solutions, 0 and 1. Meyl includes 1, but also excludes 0. This reminds me, again, of the controversy surrounding the best way to explain \(0! = 1\), since the philosophical underpinning is the same: Does it make sense to talk about the arrangement of zero things?

All three of these proofs, incidentally, are filled with proofs by contradiction (\(\neg P\) leads to a contradiction, hence \(P\)) and exhaustion (assume a value is even, then assume a value is odd). This is contrast to how we tend to teach proofs in geometry (which is what I’m currently teaching), which are nearly exclusively proofs based on known postulates and theorems.

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