Twin prime conjecture hypothesis

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.

• 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 $c$. – Mark Bennet Dec 26 '15 at 13:04
• That should be $c=6$ in last comment. – Mark Bennet Dec 26 '15 at 13:15
• * means multiply and I imposed a restriction on a and b. The question is updated. @Mark Bennet – Tony Dec 26 '15 at 13:17
• 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. – Mark Bennet Dec 26 '15 at 13:23
• @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? – Tito Piezas III Dec 26 '15 at 16:34 1 Answer 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.