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Euler's 1747 conjecture on sums of integers with restricted prime factors

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We prove that every positive integer congruent to 22 modulo 44 is a sum of two positive integers whose prime factors are all congruent to 11 modulo 44, with 11 admitted as a summand. Euler proposed this statement in a letter to Goldbach dated May 6, 1747. The proof counts representations by four squares and bounds those containing a common prime divisor congruent to 33 modulo 44 in either coordinate pair. An average over a nonsplit torus normalizer converts the quaternionic Hecke bound into an error independent of the sieving prime. Elementary estimates then give the explicit threshold 64464^4. An exact finite arithmetic certificate covers the remaining 4,194,3044{,}194{,}304 integers in the required residue class.

Provenance statement

OpenAI Codex generated the mathematical argument, performed exact arithmetic checks, and drafted the manuscript under human direction. Some parts of the contribution have been checked and understood by a human.

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References

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Version history

  1. v1 · 8 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 12, 2026