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The Lth primorial ($p_L\#$) is the product of the first L prime numbers. The reduced residue system modulo $p_L\#$ is any set of positive integers with cardinality equal to the totient of $p_L\#$ consisting only of those integers that are relatively prime with $p_L\#$. In this post, I only consider those reduced residue systems whose members are all smaller than the system's modulus (p_L#). I refer to those positive integers a and b such that $a + 2 = b$ as twins.

All prime numbers must be members of the reduced residue system modulo some primorial with the sole exception of 2, for which odd numbers are the only candidates for reduced residue system membership.

Do the reduced residue systems modulo primorials with $L > 2$ always contain twins?

NOTE: this is something of a "red herring" question, as I am really looking for sources for a paper I am writing and am wondering is anyone is aware of such a proof. Also note that, assuming the above is true, the changes to the overall patterns in reduced residue systems that each new primorial introduces go to 0 as you approach infinity, suggesting that a proof of the above would literally be infinitely close to a proof of the twin prime conjecture.

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Probably kind of late, but yes, the formula for the number of twins and quads is precisely equal to Product[Prime[z] - 2, {z, 2, i}]

See http://oeis.org/A059861

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