math.NT — Number Theory

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

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

We prove that every positive integer congruent to $2$ modulo $4$ is a sum of two positive integers whose prime factors are all congruent to $1$ modulo $4$, with $1$ 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 $3$ modulo $4$ 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 $64^4$. An exact finite arithmetic certificate covers the remaining $4{,}194{,}304$ integers in the required residue class.