Are there an infinite number of prime quadruples of the form 10 n + 1, 10 n + 3, 10 n + 7, 10 n + 9?In base 10, any prime number greater than 5 must end with the digits 1 1, 3, 7, or 9. For some n, 10 n + 1, 10 n + 3, 10 n + 7, 10 n + 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.The only progress I've been able to make is to show that n must be of the form 3 k + 1 by considering the system of modular inequalities p ≢ 0 mod 2 p ≢ 0 mod 3 p ≢ 0 mod 5

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Are there an infinite number of prime quadruples of the form   10  n  +  1,   10  n  +  3,   10  n  +  7,   10  n  +  9?In base   10, any prime number greater than 5 must end with the digits 1  1,   3,   7, or   9. For some   n,   10  n  +  1,   10  n  +  3,   10  n  +  7,   10  n  +  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.The only progress I&#039;ve been able to make is to show that   n must be of the form   3  k  +  1 by considering the system of modular inequalities  p  ≢  0    mod      2  p  ≢  0    mod      3  p  ≢  0    mod      5

Corinne Choi · Accepted Answer

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.

Are there an infinite number of prime quadruples of the form 10 n + 1 , 10 n

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