In base 10, any prime number greater than 5 must end with the digits $1$, $3$, $7$, or $9$. For some $n$, $10n + 1$, $10n + 3$, $10n + 7$, $10n + 9$ are all prime: for example, when $n=1$, we have that $11$, $13$, $17$, and $19$ are all prime. My question is, can anyone disprove the claim that there are an infinite number of such primes. (If you can prove it, that is fine, too, but seeing as it is a stronger form of the twin primes conjecture, I'd imagine that would be difficult).

The only progress I've been able to make is to show that $n$ must be of the form $3k + 1$ by considering the system of modular inequalities $$p \not\equiv 0 \mod{2}$$ $$p \not\equiv 0 \mod{3}$$ $$p \not\equiv 0 \mod{5}.$$

Beyond that, I don't know where to go.

  • $\begingroup$ Would proving a series help here? Or would you still need to prove it's a prime? $\endgroup$ – Cole Johnson Aug 3 '13 at 22:56
  • $\begingroup$ @Cole"Cole9"Johnson Could you elaborate one what you mean by "proving a series"? $\endgroup$ – user88319 Aug 3 '13 at 23:04
  • $\begingroup$ I mean that thing where you add another thing on the left and then you replace k with k+1 on the right. I think it's called induction. $\endgroup$ – Cole Johnson Aug 3 '13 at 23:07
  • $\begingroup$ @Cole"Cole9"Johnson I doubt induction would work. If I suspected a pattern existed in the $n$, and if I had some trick for proving a number was prime, I could use induction to prove the pattern worked for all $n$ (which would also prove that an infinite number of such $n$ existed). However, I don't think there's any (obvious) pattern, and I doubt there'd be any trick for proving the numbers given by the pattern were prime. $\endgroup$ – user88319 Aug 3 '13 at 23:27
  • $\begingroup$ For lists of examples, see OEIS A007811 and A173037 etc. $\endgroup$ – Jeppe Stig Nielsen Jul 23 '18 at 22:06

I suppose, you will not find a proof for neither the positive nor the negative result here.

  • The positive result would obviously imply the (unproven and presumably difficult) twin prime conjecture.

  • The negative result would disprove the first Hardy-Littlewood conjecture about the density of prime sets with a given pattern, which (among other things) conjectures a (positive) density for prime quadruples.

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    $\begingroup$ I'd hoped someone could provide a disproof, but you're probably right. (And I hadn't know about the Hardy-Littlewood conjectures, so at least I learned some neat new math). Thanks! $\endgroup$ – user88319 Aug 3 '13 at 23:29
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    $\begingroup$ Except for (3,5),(5,7), the difference between consecutive twin primes is obviously at least six, for example (5, 7) and (5+6, 7+6) = (11, 13). So this would not only prove there are infinitely many twin primes, but infinitely many pairs of twin primes at the smallest possible distance apart. $\endgroup$ – gnasher729 May 17 '14 at 19:28

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