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After reading this https://www-users.cs.york.ac.uk/~susan/cyc/p/primeprf.htm , I have question on how do we know or "see" the sense on the 4th statement, "Hence it is either prime itself, or divisible by another prime greater than $p_n$ " which meant if P is not a prime, it must be a product of higher prime?

I start by considering it cannot be any even number, as it contain constitution of "2" which is divisible by 2 which left only odd number but the odd number must be prime number?

Also in most of the proof of infinite prime $P=p_1p_2....p_n+1$ , are the proof actually meant P is not divisible by any prime in the list only(excluded higher prime like $p_{n+1}$),by definition from the list of prime use in above formula and definition of prime, P is a new prime?

Bill Dubuque
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chuackt
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    The assumption was there is no $p_{n+1}$. If $P$ is not a prime, it must be divisible by some prime, and both lead to contradictions. – player3236 Sep 12 '20 at 07:39

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No $P$ needs not to be a prime. But $P$ isn’t divisible by a prime in $\{p_1, \dots,p_n\}$. So either it is a prime or it is divisible by a prime larger than the one in the least. In either cases, it exists a prime not in the list.

mathcounterexamples.net
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  • I took a look again on the link, as an example, when P is not a prime 23571113171923293137 + 1 = 7420738134811 = 18160611*676421 , noticed 181,60611,676421 are all prime numbers is this true for all cases when P is not a prime? I don't get a clear sense on this, I'm very new to number theory – chuackt Sep 12 '20 at 09:27
  • Couldn't it be a case when P is not prime, one of the higher number that is not in the list is not a prime number – chuackt Sep 12 '20 at 09:32