# Equation representing all numbers

Joe Roberts writes, in Lure of the Integers, that Matijasevič showed that "every integer has a representation in the form $a^2+b^2+c^2+c+1$". The citation he gives is

Ju. V. Matijasevič, A Diophantine representation of the set of prime numbers (Russian), Dokl. Akad. Nauk SSSR 196 (1971) 770-773.

(Surely the intent was that all positive integers are of the indicated form, where $a,b,c$ can take on nonnegative integer values.) I was able to find an English summary of the paper

but it explained only the main result, not this (apparently) ancillary result. What is known about this result and other polynomials representing all positive integers? (Aside from all the standard four-square stuff, that is.) Is there an English-language proof, preferably simplified from the original?

If you multiply by 4 you get $$4 n = 4 a^2 + 4 b^2 + (2c+1)^2 + 3,$$ or $$4 n - 3 = 4 a^2 + 4 b^2 + (2c+1)^2 .$$ As it happens, the ternary form $$u^2 + 4 v^2 + 4 w^2$$ is regular in the sense of Dickson and does represent all numbers $\equiv 1 \pmod 4.$ So this result long predates your author, it was known by 1939 and was probably known to Gauss. See Dickson_Diagonal at ME
Finally, there are some surprises if higher degree terms are allowed. By density arguments, you would expect $x^2 + y^2 + z^9$ to integrally represent all, or all sufficiently large, positive integers, but in fact $$x^2 + y^2 + z^9 \neq 216 p^3$$ for any positive prime $p \equiv 1 \pmod 4.$