Oddmund Kolberg was probably the first to prove a related result, namely that the partition function takes both even and odd values infinitely often.
The proof employed was of elementary nature and easily accessible, and was proposed as an exercise by Newman in the American Mathematical Monthly.
[5] Ken Ono, an American mathematician, made further advances by exhibiting sufficient conditions for the conjecture to hold for prime m. He first showed that Newman's conjecture holds for prime m if for each r between 0 and m-1, there exists a nonnegative integer n such that the following holds: He used the result, together with a computer program, to prove the conjecture for all primes less than 1000 (except 3).
[6] Ahlgren expanded on his result to show that Ono's condition is, in fact, true for all composite numbers coprime to 6.
[8] Afterwards, Ahlgren and Boylan used Ono's criterion to extend Newman's conjecture to all primes except possibly 3.
has at least 1 solution has been proved for all m. It was formerly known as the Erdős–Ivić conjecture, named after mathematicians Paul Erdős and Aleksandar Ivić.