On Euler's Theorem

TL;DR

This paper explores and proves new equalities among various partition functions related to Euler's theorem, extending classical results by establishing novel relationships between different partition counts.

Contribution

It introduces new identities linking partition functions involving distinct, odd, and constrained parts, expanding the understanding of partition equivalences.

Findings

Proves that for n>0, A(n)=B(n)=C(n+1)=D(n+1)/2.

Establishes new equalities among partition counts with specific constraints.

Extends classical Euler's theorem to more complex partition functions.

Abstract

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.