The numbers that are not of the form x2 + y2 + 10z2 are 4λ(16μ + 6).

even

The numbers that are not of the form x2 + y2 + 10z2, viz. 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, ... do not seem to obey any simple law.

odd

In his 1916 paper[3] Ramanujan made the following observations about the form x2 + y2 + 10z2.

Every integer of the form 10n + 5 is represented by Ramanujan's ternary quadratic form.

If n is an odd integer which is not then it can be represented in the form x2 + y2 + 10z2.

square-free

There are only a finite number of odd integers which cannot be represented in the form x2 + y2 + 10z2.

If the is true, then the conjecture of Ono and Soundararajan is also true.

generalized Riemann hypothesis

Ramanujan's ternary quadratic form is not regular in the sense of .[5]

L.E. Dickson

The conjecture of Ken Ono and Soundararajan has not been fully resolved. However, besides the results enunciated by Ramanujan, a few more general results about the form have been established. The proofs of some of them are quite simple while those of the others involve quite complicated concepts and arguments.[1]