Proving it is impossible; on Erd\H{o}s problem $\# 278$

TL;DR

This paper investigates Erdős problem 78, showing it likely has no general solution by linking it to a complex knapsack problem and revealing structural dependencies affecting formula equivalence.

Contribution

It establishes a connection between Erds problem 78 and hard knapsack instances, providing evidence of the problem's intractability and structural complexity.

Findings

Different formulas emerge based on the structure of the moduli.

The problem appears to have no general solution due to its relation to complex knapsack instances.

Structural variations significantly influence the problem's solvability.

Abstract

Erd\H{o}s asked many mathematical questions. Some lead to exciting research, others turned out to be easily solved. In this article, we provide evidence that one of his questions, Erd\H{o}s problem \#278 , has no general answer. We do so by relating it with a hard knapsack problem instance,and by demonstrating that different, non-equivalent formulas arise depending on the structure of the moduli.