1

The question kinda tell everything for itself,

let P(n) be the n-th prime number, is

$(\Pi_1^n P_n)+1$

prime ?

Ludovic Zenohate Lagouardette
  • 11

2 Answers2

5

No, consider $1 + 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19$. It's divisible by $347$ and $27953$.

Ashvin Swaminathan
  • 2,706
  • The composite ones are given in http://oeis.org/A066576, starting with 30031, 510511, 9699691, 223092871, 6469693231, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271 – Ross Millikan Jul 14 '16 at 22:29
0
> gp
                            GP/PARI CALCULATOR Version 2.7.3 (released)
                    i386 running darwin (x86-64/GMP-6.0.0 kernel) 64-bit version
       compiled: May 24 2015, Apple LLVM version 6.0 (clang-600.0.57) (based on LLVM 3.5svn)
                                      threading engine: single
                           (readline v6.3 enabled, extended help enabled)

                               Copyright (C) 2000-2015 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY
WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000
? factor(2*3*5*7*11*13*17*19+1)
%1 =
[  347 1]

[27953 1]

Some people maintain that Euclid proved the infinitude of primes by contradiction, but actually he made a direct proof.

Some proofs by contradiction run as follows.

Suppose $p_1,p_2,\dots,p_n$ are all the primes. Then $N=p_1p_2\dotsm p_n+1$ is not divisible by any of the given primes, so it is prime. Contradiction.

However this doesn't prove that the product of consecutive primes plus one is prime.

egreg
  • 238,574
  • If p1, p2, ..., pn were a complete list of all primes, then N would indeed be a prime. But it isn't a complete list of all primes. – gnasher729 Jul 14 '16 at 22:36
  • @gnasher729 Exactly; the conclusion that $N$ is prime is a byproduct of a false assumption and we know that “false implies anything”. – egreg Jul 14 '16 at 22:37