## S000118

Lists of four numbers (a,b,c,d) such that a*x^2 + b*y^2 + c*z^2 + d*u^2 represents all nonnegative numbers.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 2, 3, 1, 1, 3, 3, 1, 2, 3, 3, 1, 1, 1, 4, 1, 1, 2, 4, 1, 2, 2, 4, 1, 1, 3, 4, 1, 2, 3, 4, 1, 2, 4, 4, 1, 1, 1, 5, 1, 1, 2, 5, 1, 2, 2, 5, 1, 1, 3, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 1, 1, 6

1

We have a <= b <= c <= d. The lists of four numbers are sorted. Ramanujan listed 55 quadruples, but (1,2,5,5) does not belong because it does not represent 15. Dickson discovered this error in 1927. - T. D. Noe, Jul 08 2014

T. D. Noe, Plot of 54 * 4 terms

T. D. Noe, Table of 54 * 4 terms

L. E. Dickson, Integers represented by positive ternary quadratic forms, Bulletin of the American Math. Soc., vol XXXIII (1927), pp. 63-70.

Srinivasa Ramanujan, On the expression of a number in the form ax^2 + by^2 + cz^2 + du^2, Proc. Camb. Phil. Soc. 19 (1916), pp. 11–21 and Collected Papers, pp. 169-178.

(Mma) Monitor[nn = 10; mx = 20; t = {}; Do[u = Union[Flatten[Table[a*x^2 + b*y^2 + c*z^2 + d*u^2, {x, 0, nn}, {y, 0, nn}, {z, 0, nn}, {u, 0, nn}]]]; If[Complement[Range[nn^2], u] == {}, AppendTo[t, {a, b, c, d}]], {a, 1}, {b, mx}, {c, b, mx}, {d, c, mx}]; t, {a, b, c, d}]; t

Cf. S000122, S000123.

nonn,fini,full,nice

T. D. Noe, Jul 02 2014