Editing Quadratic form (section)

=== Universal quadratic forms ===
An integral quadratic form whose image consists of all the positive integers is sometimes called ''universal''.  [[Lagrange's four-square theorem]] shows that {{math|''w''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>}} is universal. [[Ramanujan]] generalized this {{math|''aw''<sup>2</sup> + ''bx''<sup>2</sup> + ''cy''<sup>2</sup> + ''dz''<sup>2</sup>}} and found 54 multisets {{math|{{mset|''a'', ''b'', ''c'', ''d''}}}} that can each generate all positive integers, namely,
{{plainlist | indent = 1 |
* {{math|{{mset|1, 1, 1, ''d''}}, 1 ≤ ''d'' ≤ 7}}
* {{math|{{mset|1, 1, 2, ''d''}}, 2 ≤ ''d'' ≤ 14}}
* {{math|{{mset|1, 1, 3, ''d''}}, 3 ≤ ''d'' ≤ 6}}
* {{math|{{mset|1, 2, 2, ''d''}}, 2 ≤ ''d'' ≤ 7}}
* {{math|{{mset|1, 2, 3, ''d''}}, 3 ≤ ''d'' ≤ 10}}
* {{math|{{mset|1, 2, 4, ''d''}}, 4 ≤ ''d'' ≤ 14}}
* {{math|{{mset|1, 2, 5, ''d''}}, 6 ≤ ''d'' ≤ 10}}
}}
There are also forms whose image consists of all but one of the positive integers.  For example, {{math|{{mset|1, 2, 5, 5}}}} has 15 as the exception.  Recently, the [[15 and 290 theorems]] have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.