Are there an infinite number of prime quadruples of the form $10n + 1$, $10n + 3$, $10n + 7$, $10n + 9$?

prime-numbers elementary-number-theory

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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