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In the proof for the existence of unlimited prime numbers, i saw the following

let n be the number of prime numbers as P1,P2,P3,.......Pn
let a = P1P2P3....Pn+1
a > Pn and a is not a prime number
a should have a prime factor
let P be it
P|a ----> 1
but P|a-1 ----->2
1 & 2 -> P|1
but 1 doesn't have a factor ........
and the proof follows

but i didn't understand the symbol P|a
what does P|a means? please somebody explain me

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3 Answers 3

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It means $P$ is a divisor of $a$.

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  • $\begingroup$ if its the divisor of a then how do and a-1 have the same divisor $\endgroup$ Commented Jul 25, 2015 at 12:21
  • $\begingroup$ Because $a-1$ is the product of all primes. $\endgroup$ Commented Jul 25, 2015 at 12:24
  • $\begingroup$ a is the product of all prime numbers no $\endgroup$ Commented Jul 25, 2015 at 12:25
  • $\begingroup$ Of course $a$ is not. $a-1$ is. $\endgroup$ Commented Jul 25, 2015 at 12:30
  • $\begingroup$ but we assume a to be the multiplication of prime numbers $\endgroup$ Commented Jul 25, 2015 at 12:35
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P|a means P divides a.For P|a we can also write this as Pc=a where c is a constant.It simply means that $$P*c=a$$ or P is a factor of a.

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It means $p$ is a divisor of $a$, i.e $a$ leaves a remainder of $0$ when divided by $p$. You can also write $a \equiv 0 \text{ mod }p$.

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