The form x^2 + y^2 + 10*z^2 (x, y, z integers) is known as Ramanujan's ternary quadratic form after appearing in a 1916 paper of Ramanujan. There is a known formula producing all even natural numbers not representable in this form, but there are only 18 known odd natural numbers not representable (OEIS sequence A003585):
- 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719
In 1997 K. Ono and K. Soundararajan proved (Ono, Soundararajan, "Ramanujan's Termary Quadratic Form") that assuming the generalized Riemann hypothesis (GRH) that this list is complete, i.e. every odd integer above 2719 is representable in this form. At the time W. Galway verified this for all odd integers between 2719 and 2e+10, I have currently verified this up to 7.5e+10. rtqf.c searches for a counterexample/a non-representable odd integer above this, if one is found this would disprove GRH. (This would not necessarily disprove RH, so finding a counterexample does not guarantee you win a $1,000,000 award.)