A summary of Euler's work on the pentagonal number theorem
The author gives a very detailed mathematical as well as historical case study of Euler's work on the pentagonal number Theorem \[ (1-x)(1-x^2)(1-x^3) \dots = 1-x-x^2+x^5+x^7+\dots \] (the exponents of the infinite series are the pentagonal numbers \(\omega_n = n(3n-1)/2\), \(n = 0, \pm1, \pm2,\dots)\), this is one of the most important results of Euler. He himself spoke of the ``sehr wunderbaren Ordnung in den Zahlen''. This equality inductively found by Euler about in autumn 1740 was stated but not proved yet by Euler in ``Observatio analyticae\dots'' (E150), read before the Academy on May 15, 1741 and published in the Academy journal ``Commentarii'' in 1751 (for 1741). Bell's paper describes in detail how Euler found a general proof. For this purpose the topic is embedded in a glimpse of the prehistory, in Euler's related correspondence (above all with Christian Goldbach) and an unpublished proof (Euler's notebooks). The paper begins with Euler's work on infinite products (for example \(\sin z\), \(\zeta(s)\)) and some related infinite series. The author regards the history of the theorem as ``an excellent case for understanding the relation to Euler between examples of a theorem that seem like they could be generalized and a general proof of the theorem'' (p. 303). The Theorem was stated in some papers (E158, E 175 and E243) before Euler finally was able to prove the Theorem in ``Demonstratio theorematis\dots'' (E244) in the ``Novi Commentarii'' for 1754/55 (printed in 1760); some letters to Goldbach anticipated the papers (21. 3. 1747, 29. 5. 1750); in \S3 Bell discusses also a more complicated unpublished proof contained in Euler's Note books. Euler's last papers on the Theorem and pentagonal numbers were written 1775 (E541, E542). In the beginning Euler inferred the Theorem from the recurrence formula \(\sigma(n)\) for the sum of divisors of a natural number \(n\), where by a daring incomplete induction he justified the formula for \(\sigma(n)\). The deduction of theorems by concrete numerical material (incomplete induction) served as heuristic device. However, it can indicate -- at least for a genius like Euler -- how a proof could work. Applications to divisor sums and integer partitions are given attention, and some related results in the theory of analytic functions are included. The author states that ``Euler had a good idea of what a proof was'' (p. 370). Indeed, the paper shows that Euler had a good feeling for and understanding of the inductive method and how to use limited concrete numerical material for mathematical induction. However, the general expression ``rigorous proof'' (p. 370) with Euler would deserve a longer discussion of the nature of a proof in the 18th century.
scientific article; zbMATH DE number 5790092
scientific article