Every integer greater than 454 is the sum of at most seven positive cubes

TL;DR

This paper proves a long-standing conjecture that all positive integers greater than 454 can be expressed as the sum of at most seven positive cubes, confirming a hypothesis first observed in the 19th century.

Contribution

The authors complete the proof of the conjecture, building on previous mathematical work and computational efforts to establish the result for all integers above 454.

Findings

All integers greater than 454 are sums of at most seven positive cubes.

The conjecture is now fully proven, resolving a 19th-century mathematical question.

The proof consolidates previous partial results and computational verifications.

Abstract

A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramar\'e, Boklan, Elkies, and many others.