Are there infinitely many nonnegative integers not of the following four forms? The four forms are:
$3x^2 + (6y-4)x - y$
$3x^2 + (6y-2)x + y - 1$
$3x^2 + (6y-3)x + y - 1$
$3x^2 + (6y-3)x - y\ $,$\ x,y \in \mathbb{Z}^{+}$
For example: $4=3 \cdot1^2+(6 \cdot1-4) \cdot 1-1\ $ is the minimum number of the four forms, so that $\ 0, \ 1, \ 2,\ 3\ $  must not be of the four forms. How can one prove that the set of these numbers is infinite?
 A: Let me denote these forms by $f_i(x,y)$ for $i=1,2,3,4$, respectively. Notice that
$$12\cdot f_1(x,y) = (6x - 1)\cdot (12y + 6x - 7) - 7\\
12\cdot f_2(x,y) = (6x + 1)\cdot (12y + 6x - 5) - 7\\
12\cdot f_3(x,y) = (6x + 1)\cdot (12y + 6x - 7) - 5\\
12\cdot f_4(x,y) = (6x - 1)\cdot (12y + 6x - 5) - 5$$
Now, if $(12k+5, 12k+7)$ is a twin prime pair, then $k$ is not represented by either of these four forms with positive $x,y$.
P.S. Suprisingly, this question is very similar to the one I answered yesterday: at https://mathoverflow.net/questions/164792/are-there-infinitely-many-natural-numbers-not-covered-by-one-of-these-7-polynomi/164859#164859
A: Here is a solution without assuming (a stronger form of) the twin prime conjecture. This paper by Iwaniec shows that if $P(x,y)=ax^2+bxy+cy^2+ex+fy+g$ satisfies some mild conditions that are easy to check and valid in this case, then the number of primes up to N represented by $P(x,y)$ is $\ll \frac{N}{(\log N)^{\frac{3}{2}}}$, so the quadratic polynomials in the question represent just a zero density set of the primes, and the same argument applies to any such finite collection quadratic polynomials as long as the polynomials satisfy the quite simple requirements of theorem 1 of Iwaniec's paper. The conditions are:


*

*$\frac{dP}{dx}$ and $\frac{dP}{dy}$ are linearly independent,

*$P(x,y)$ is irreducible in $\mathbb{Q}[x,y]$,

*$b^2-4ac$ not a perfect square,

*$gcd(a,b,c,d,e,f,g)=1$,

*$af^2-bef+ce^2+(b^2-4ac)g\neq 0$

*$P(x,y)$ represents arbitrarily large odd integers.


EDIT: I was too hasty; in this case $b^2-4ac=b^2$ is actually a perfect square. Then Iwaniec's paper actually says that the number of primes up to $N$ represented by $P(x,y)$ is $\gg \frac{N}{\log N}$, so this does not work (unless there is some way to estimate the constant).
