1

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 $C = \{3,5,7,10,…\}$.

Suzet
  • 6,055
busy Ang
  • 51
  • 1
    https://math.stackexchange.com/questions/2496106/diophantine-equation-with-application-to-twin-primes and https://math.stackexchange.com/questions/2324324/proof-of-minor-claim-related-to-the-twin-primes-conjecture and https://math.stackexchange.com/questions/550916/find-all-integers-n-such-that-n-neq-6xy-pm-x-pm-y and https://math.stackexchange.com/questions/1416570/largest-number-that-cannot-be-expressed-as-6nm-n-m and https://math.stackexchange.com/questions/698021/does-proving-the-following-statement-equate-to-proving-the-twin-prime-conjecture – Gerry Myerson Aug 05 '18 at 03:40

1 Answers1

1

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.

Somos
  • 37,457
  • 3
  • 35
  • 85