Around Euler's theorem on sums of divisors

V. M. Zyuz'kov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 51 (2018), pp. 19-32

Résumé

This work relates to experimental mathematics. Two problems solved by Euler are considered. In the first task, the number of partitions for natural numbers is counted; the solution of the second task gives the recursion regularity connecting the sums of dividers of natural numbers. Euler had no definition of the formal ascending power series and a generating function; nevertheless, using the inductive reasonings, he obtained results which were rigorously proved later by other mathematicians. The paper shows how to solve these problems by means of the apparatus of generating functions and calculations in the Mathematica system. Solving of these tasks, Euler considered two infinite sequences, $\{a_n\}_{n=0}^\infty$: $1, -1, -1, 0, 0, 1, 0, 1, 0, 0, \dots$ and $\{b_n\}_{n=0}^\infty$: $1, 2, 5, 7, 12, 15, 22, 26, \dots$. However, the author has obtained new results: a “closed form” for these sequences and a generating function for the sequence $\{b_n\}_{n=0}^\infty$.

Détail

Keywords: experimental mathematics, Euler's theorem of partitions, Euler's hypothesis of the sums of dividers, generating functions, Mathematica system.