of quadratic forms, the 'local' source of the difficulty; the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s; the actual needs for integral quadratic form theory in topology for intersection theory; the Lie group and algebraic group aspects. 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 is universal. Ramanujan generalized this to and found 54 multisets that can each generate all positive integers, namely, {1, 1, 1, d}, 1 ≤ d ≤ 7 {1, 1, 2, d}, 2 ≤ d ≤ 14 {1, 1, 3, d}, 3 ≤ d ≤ 6 {1, 2, 2, d}, 2 ≤ d ≤ 7 {1, 2, 3, d}, 3 ≤ d ≤ 10 {1, 2, 4, d}, 4 ≤ d ≤ 14 {1, 2, 5, d}, 6 ≤ d ≤ 10 There are also forms whose image consists of all but one of the positive integers. For example, {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. See also ε-quadratic form Cubic form Discriminant of a quadratic form Hasse–Minkowski theorem Quadric Ramanujan's ternary quadratic form Square class Witt group Witt's theorem Notes References Further reading External links Linear algebra Real algebraic geometry Squares in number theory