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The primorials are the KEY to locating any and all prime numbers. One can easily convince oneself of the requirement being a prime offset from a primorial as a necessary (but not sufficient) condition for a prime number. — Bill McEachen 04:40, 18 December 2010 (UTC)

Obviously, this prime offset must be coprime to the primorial number in question!
A composite offset that is coprime to the primorial number in question may give a prime number, e.g.
2*3*5*7 = 210, 210 - 11^2 = 210 - 121 = 89 a prime number!
So the necessary (but not sufficient) requirement is that the offset be coprime to the primorial in question. — Daniel Forgues 05:49, 18 December 2010 (UTC)
Well, I understand your comment, but the offset has to exist to meet any requirement. — TomAto, Tomato Bill McEachen 21:57, 23 December 2010 (UTC)
If the offset doesn't exist, then it is 0, and GCD(0, n) = n for all n > 0, so GCD(0, n) ≠ 1 for all n ≠ 1 (not coprime) and 0 is only coprime to 1 (all integers being coprime to 1.) Although 1 was considered prime in the past, it is not so anymore, 1 is a unit (has a multiplicative inverse,) the empty product of primes. — PotAto, Potato Daniel Forgues 02:52, 24 December 2010 (UTC)
This is great! from a post a few years ago on Wikipedia talk page, we have "...Proving true conjectures can be hard or almost impossible, and I see no strong reason to believe your conjecture is true. Based on heuristics each prime appears to have a tiny chance of being a counter example but there are infinitely many primes. If a conjecture hasn't been proved then it's useless to prove other things such as whether a number is composite. And if your conjecture was actually proved (which I think nobody would have a clue how to approach)..."
This comment from Jens Anderson (aka PrimeHunter) I think, so your feedback makes the conjecture seem trivial to prove. I am definitely not a proof guy. I very much appreciate the insight. So, for clarity, are you saying the offset trivially exists for each prime or ?? (You must be patient with me). The part I like(ed) is generating larger primes from smaller ones, as in your example above of having used 2,3,5,7 and 11 to get 89. When I first noticed this the inherent structure to how they all relate was fascinating to me, having heard nothing but how "arbitrary" they are.
If the offset is not coprime to the primorial in question (i.e. GCD(primorial, offset) > 1) then GCD(primorial, offset) being a divisor (greater than 1) of both primorial and offset, it is obviously a divisor of primorial - offset and primorial + offset, thus both are composite. QED — Daniel Forgues 04:46, 27 December 2010 (UTC)


By analogy with the multifactorial (double factorial, triple factorial, ...) is there any point in having multiprimorial (double primorial, triple primorial, ...) or is it useless... — Daniel Forgues 05:47, 1 October 2012 (UTC)