I have a bivariate integer function where x,y are positive integers in the function $f(x,y)=5+23x+7y+30xy$. The lattice points of this function, or its range, contain a large number of values. I'm trying to see if there is any way to create a positive, increasing, infinite sequence or function which avoids the range of f. Or a recursive sequence whose values avoid the range of f.



Avoiding the range of a bivariate function or Diophantine function
By [Dirichlet's theorem](https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions), there are infinitely many positive integers $k$ such that $30k+11$ is a prime. I claim that no such $k$ is in the range of your function. Indeed, if we had $k=5+23x+7y+30xy$ for some positive integers $x$ and $y$, then we also had
$30k+11=(7+30x)(23+30y)$, a contradiction.


