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This question is a follow up of this-one - where in I'm going through a youtube tutorial (https://www.youtube.com/watch?v=pNhbW1Hrjcs) and at 7:31 in the video.

Basically to prove, we consider a primorial(n) where n is the index in the set of primes and then we add one to it and we say that if primorial(n)+1 is prime then point proved and if its composite then we have a prime P(i) at index 'i' such that it divides the primorial and then we proceed saying that we also have 1/P(i) which cannot be a natural number (that is to say it would be a fraction). We conclude the proof on basis of the previous point.

However as discussed in the previous question, and quoting lulu -

The first n primes all divide primorial(n). Hence none of them can divide primorial(n)+1.

Now, the above point (which I believe to be correct and self-explanatory and hence the question) implies that our technique/approach for proving the primality is incorrect.

Is my understanding correct?

Ravindra HV
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    No, your understanding is faulty. We can have a prime that divides primorial($n$), but it obviously won't be one of the first $n$ primes. – TonyK Jul 25 '21 at 22:23
  • Ok. So then the proof is on the lines that - if 'primorial(n)+1' is prime then point proved and if its composite, even then, the factor would be outside of the set of primes used to construct the primorial ? That does make sense. Can you post that as an answer? Will accept that. – Ravindra HV Jul 25 '21 at 22:33
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    I think you meant natural number when you typed real number – J. W. Tanner Jul 25 '21 at 22:38
  • @J.W.Tanner - Yes. Thanks! Will correct it ! – Ravindra HV Jul 25 '21 at 22:46

2 Answers2

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Note - From TonyK's comment to the post and to quote

No, your understanding is faulty. We can have a prime that divides primorial(n), but it obviously won't be one of the first n primes.

and also from this answer for a similar question (and to quote)

....Suppose that p1,...,pt are all the primes and consider the number N=p1⋯pt+1. The number N must be divisible by some prime (possibly itself, but this is irrelevant for the argument) but since noone of the pi divides N, this gives a contradiction.

it would appear that the approach to prove primality in the video is not correct.

Will leave this answer as a placeholder until there is an answer that I can accept!

Ravindra HV
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  • I think you are confused about what the video tries to prove. It does not try to prove that $q=p_n# +1$ is necessarily prime. It proves that $q=p_n# +1$ is greater than any prime and hence composite by the assumption that all primes are contained in a set of numbers $\le \max(p_n)$. Being composite, $q$ must be a product of primes. Any prime (the video calls it $p_1$ for convenience) which divides $q$ must also divide $p_n# +1$, which forces $\frac{1}{p_1}$ to be an integer. Since this is impossible, the basic assumption that the primes are finite in number must be wrong. – Keith Backman Jul 26 '21 at 01:16
  • In the video, the author is dividing p1 with the corresponding entry in the set of primorials. But from the discussion (refer previous question I referenced in the beginning of the post), it’s clear that the factors that divide {p(n)#+1} cannot be part of the set that constitutes the primorial. That is what made me think that the approach in the video is incorrect. – Ravindra HV Jul 27 '21 at 05:49
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    You're almost there: The facts that (a) $p_n#+1$ must have prime factors and (b) the prime factors that divide $p_n#+1$ cannot be factors of $p_n#$ forces some conclusion. You suggest that perhaps the definition of a prime is incorrect, or the methodology in the video is incorrect, but the other possible conclusion is that the assumption made originally, that there are a finite number of primes, is what is incorrect. There is no way to reconcile your suggested conclusions with mathematics, but the conclusion that primes are infinite in number does fit with mathematics. – Keith Backman Jul 27 '21 at 15:29
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Here's how the method properly works:

  1. we have a set of primes we assume are all primes, say $2,3,5,7,11,13$
  2. we multiply these together, and add a number none of them divide ( namely 1)
  3. we then reach a contradiction of our first step,

this new number is greater than 1, so it is either composite ( having prime factors but none of which could be in the original set, as factoring out those out of 1 is impossible in the integers) , or prime ( but it wasn't in the starting set). Therefore, it follows our starting set is incomplete ( we could add any primes we find anew, and repeat to find more).

Roddy MacPhee
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  • Yes. This is the understanding I have and this answer would basically mean that my understanding is correct. And consequently the approach in the video at the very least needs to be improved upon to include above facts (that's what I believe). – Ravindra HV Aug 01 '21 at 13:41