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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 $C = \{3,5,7,10,…\}$.

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closed as off-topic by Xander Henderson, Jendrik Stelzner, Brahadeesh, José Carlos Santos, Taroccoesbrocco Aug 6 '18 at 8:21

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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.

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  • $\begingroup$ thank you very much for the feedback. $\endgroup$ – busy Ang Aug 5 '18 at 4:51

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