Prime numbers as a uniqueness set of the parallelogram equation via the Goldbach's conjecture

TL;DR

This paper explores the relationship between prime numbers, the parallelogram functional equation, and Goldbach's conjecture, revealing that under certain conditions, the only solution is quadratic.

Contribution

It demonstrates that multiplicative arithmetic functions satisfying the parallelogram equation on primes are uniquely quadratic, assuming Goldbach's conjecture.

Findings

The unique solution is quadratic under the given conditions.

Goldbach's conjecture plays a key role in deriving the result.

The study links prime number properties with functional equations.

Abstract

Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.