290 theorem

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. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000.

In 2005, Bhargava and Jonathan P. Hanke announced a proof of Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290. The proof is to appear in Inventiones Mathematicae.

In simple terms, the results are as follows. Suppose ${\displaystyle Q_{ij}}$ is a symmetric square matrix with real entries. For any vector ${\displaystyle x}$ with integer components, define

This function is called a quadratic form. We say ${\displaystyle Q}$ is positive definite if ${\displaystyle Q(x)>0}$ whenever ${\displaystyle x\neq 0}$. If ${\displaystyle Q(x)}$ is always an integer, we call the function ${\displaystyle Q}$ an integral quadratic form.

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