I would like to classify the integers $m \geq 2$ for which the four quadratic polynomials
$3k^2$, $3k^2+2k$, $3k^2+3k+1$, and $3k^2+5k+2$ together represent all integers modulo $m$. That is, every integer modulo $m$ should be in the range of at least one of these polynomials (where all operations are carried out modulo $m$). Computer evidence suggests that this holds if and only if $m$ is one of the following:
$7, 10, 19, 2^j, 3^j, 5^j, 11^j, 13^j, 41^j, 2\cdot3^j, 5\cdot3^j$, where $j \geq 1$.


Does someone see how to prove this? Thank you.



Four polynomials representing all integers modulo m
For a prime $p>2$, fix a nonsquare $c$. If you find $y$ such that $y/3$ is a non-square (i.e. $y/3=cx^2, x\ne0$) and $y/3 - 1/9 = cz^2, z\ne 0$, then $y$ is not represented by the first two polynomials and I can't be bothered completing the square to write the conditions for the other two. Bottom line is, you find such a $y$ if you can find a point on a curve over the finite field 
$\mathbb{F}\_p$. By Weil, this will happen as soon as $p$ is large enough. So your $m$ can only have prime factors from a finite set. Should be downhill from here.


