How can we show that there are infinitely many integers $C$ such that the simple hyperbolic diophantine equation: $$6xy ± x ± y = C$$ gives a non-integer solutions for $x, y$, except at $(0, ±C), (±C,0)$?
Some of these values of $C = \{3,5,7,10,…\}$.
This is OEIS sequence A002822 "Numbers n such that 6n-1, 6n+1 are twin primes". Thus, proving there are infinitely many of these numbers is equivalent to proving the twin prime conjecture.