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# Talk:Primorial

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## Backbone

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)

- This is great! from a post a few years ago on Wikipedia talk page, we have "...

- 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)

- 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)

## Multiprimorials

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)