Numbers in the base e^π

2025/09/19 by Simon Plouffe, Plouffe, Simon
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary 11Y40 #Secondary 11Y55

paper · pdf · doi:10.48550/arxiv.2509.15609

openalex publication_date 2025/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A large-scale experiment was conducted to find formulas relating to the base e^π. The numbers in this base are x = ∑_n=0^∞ a(n)\over e^πn where a(n) is taken from the OEIS catalog. These experiments were inspired by several facts. Indeed, it is known that the formula generating the partitions of integers is generated by an infinite product ∏_k≥1^∞ 1\over 1-x^k = ∑_n=0^∞ p(n)xⁿ that when evaluated at x=e^-π is equal to 2^3/8 Γ(3/4) \over π^1/4 e^π/24 . (1) By analyzing the 387500 sequences of the OEIS catalog, the model that was used is based on the fact that the infinite sum evaluated at e^π, is an expression that can be detected using a program like lindep from Pari-Gp. The process made it possible to find 793 expresssions similar to (1).

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