[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.
with integer components, define This function is called a quadratic form.
The 15 theorem says that a quadratic form with integer matrix is universal if it takes the numbers from 1 to 15 as values.
A more precise version says that, if a positive definite quadratic form with integral matrix takes the values 1, 2, 3, 5, 6, 7, 10, 14, 15 (sequence A030050 in the OEIS), then it takes all positive integers as values.
For example, the quadratic form is universal, because every positive integer can be written as a sum of 4 squares, by Lagrange's four-square theorem.
By the 15 theorem, to verify this, it is sufficient to check that every positive integer up to 15 is a sum of 4 squares.
The 290 theorem says a positive definite integral quadratic form is universal if it takes the numbers from 1 to 290 as values.
A more precise version states that, if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290 (sequence A030051 in the OEIS), then it represents all positive integers, and for each of these 29 numbers, there is such a quadratic form representing all other 28 positive integers with the exception of this one number.
Bhargava has found analogous criteria for a quadratic form with integral matrix to represent all primes (the set {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73} (sequence A154363 in the OEIS)) and for such a quadratic form to represent all positive odd integers (the set {1, 3, 5, 7, 11, 15, 33} (sequence A116582 in the OEIS)).
Expository accounts of these results have been written by Hahn[5] and Moon (who provides proofs).