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Let $c$ be a positive integer and fix $a=c-1$, and $b=c+1$.

Challenge: Find the largest value of $c$ such that $ac\pm1$ and $bc\pm1$ are pairs of twin primes.

For example, with $c=6$ we have $a=5$ and $b=7$ yielding twin primes $5\cdot6\pm1$ (29 and 31) and $6\cdot7\pm1$ (41 and 43).

My conjecture is that if an upper bound for $c$ can be proven, then the twin prime conjecture is false, and if it can be proven that $c$ can be arbitrarily large then an infinite number of twin prime pairs can be generated and thus the twin prime conjecture is true.

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    $\begingroup$ It is very unclear what you are asking here - what do you mean by $*$ and by square brackets? It is elementary, for example, that any twin primes other than the pair $3$ and $5$ are of the form $6n\pm 1$. So if I read you right, $b=6$ will give lots of solutions unless you put constraints on $a$ and $c$. $\endgroup$ – Mark Bennet Dec 26 '15 at 13:04
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    $\begingroup$ That should be $c=6$ in last comment. $\endgroup$ – Mark Bennet Dec 26 '15 at 13:15
  • $\begingroup$ * means multiply and I imposed a restriction on a and b. The question is updated. @Mark Bennet $\endgroup$ – Tony Dec 26 '15 at 13:17
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    $\begingroup$ So it looks like you are looking for cases where $c^2-c-1; c^2-c+1; c^2+c-1; c^2+c+1$ are all prime and the highest value of $c$ for which this happens, if there is such a highest value. $\endgroup$ – Mark Bennet Dec 26 '15 at 13:23
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    $\begingroup$ @lurker: A formatting tip: If you want to Latexify, then only one $ at each end. If you want to center it, then two $$ at each end. Doesn't your post look much neater now, by getting rid of unnecessary centering? $\endgroup$ – Tito Piezas III Dec 26 '15 at 16:34
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Of course if $c$ can be taken arbitrarily large than the twin prime conjecture is true. If there are only finitely many such $c$ then the twin prime conjecture might be true or false. But the smart money says that there are infinitely many such $(a,b,c).$ To support this claim, some examples (giving only $ac-1$):

  • 10^10 + 58244
  • 10^20 + 2025560
  • 10^30 + 16004624
  • 10^40 + 53123060
  • 10^50 + 31173119
  • 10^60 + 199288424
  • 10^70 + 186691829
  • 10^80 + 446536985
  • 10^90 + 615513365
  • 10^100 + 2863537016
  • 10^110 + 1189720544
  • 10^120 + 1662005540
  • 10^130 + 5328042449

Standard number-theoretic conjectures suggest that there are infinitely many.

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