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 arising from an integer matrix represents all positive integers up to 15, then it represents all positive integers.

Conway and Schneeberger chose not to publish their proof because Manjul Bhargava found a simpler proof, published in 2000.

Conway conjectured an analogous statement for integral quadratic forms, with the constant 15 replaced by 290.

Bhargava and Jonathan Hanke have a 2011 preprint

with a proof of this "290 conjecture", but as of 2026 it has not been published, and the code containing the computations needed for the proof is no longer available at the URL listed in their preprint.

Details

Suppose

Q

i

j

{\displaystyle Q_{ij}}

is a symmetric matrix with real entries. For any vector

x

{\displaystyle x}

with integer components, define

Q

(

x

)

=

x

t

Q

x

=

∑

i

,

j

x

i

Q

i

j

x

j

{\displaystyle Q(x)=x^{t}Qx=\sum _{i,j}x_{i}Q_{ij}x_{j}}

This function is called a quadratic form. We say

Q

{\displaystyle Q}

is positive definite if

Q

(

x

)

>

0

{\displaystyle Q(x)>0}

whenever

x

≠

0

{\displaystyle x\neq 0}

. If

Q

(

x

)

{\displaystyle Q(x)}

is always an integer, we call the function

Q

{\displaystyle Q}

an integral quadratic form.

We get an integral quadratic form whenever the matrix entries

Q

i

j

{\displaystyle Q_{ij}}

are integers; then

Q

{\displaystyle Q}

is said to have integer matrix. However,

Q

{\displaystyle Q}

will still be an integral quadratic form if the off-diagonal entries

Q

i

j

{\displaystyle Q_{ij}}

are integers divided by 2, while the diagonal entries are integers.