15 and 290 theorems

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]

  1. ^ Conway, J.H. (2000). "Universal quadratic forms and the fifteen theorem". Quadratic forms and their applications (Dublin, 1999) (PDF). Contemp. Math. Vol. 272. Providence, RI: Amer. Math. Soc. pp. 23–26. ISBN 0-8218-2779-0. Zbl 0987.11026.
  2. ^ Bhargava, Manjul (2000). "On the Conway–Schneeberger fifteen theorem". Quadratic forms and their applications (Dublin, 1999) (PDF). Contemp. Math. Vol. 272. Providence, RI: Amer. Math. Soc. pp. 27–37. ISBN 0-8218-2779-0. MR 1803359. Zbl 0987.11027.
  3. ^ Alladi, Krishnaswami. "Ramanujan's legacy: the work of the SASTRA prize winners". Philosophical Transactions of the Royal Society A. The Royal Society Publishing. Retrieved 4 February 2020.
  4. ^ Bhargava, M., & Hanke, J., Universal quadratic forms and the 290-theorem.