15 and 290 theorems
On when an integer positive definite quadratic form represents all positive integers / From Wikipedia, the free encyclopedia
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers.[1] The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000.[2]
Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290.[3] The proof has since appeared in preprint form.[4]