Find primes $p_1,p_2,..,p_6$ such that $1+\prod_{i=1}^{6}p_i $is not prime

Question

Show that if $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ and $p_6$ are primes, then $$1+\prod_{i=1}^{6}p_i$$ is not necessarily prime by using a specic example.

Answer 1

As all primes$>3$ of the form $\equiv\pm1\pmod6$

Choose $p_1,p_2,p_3\equiv-1\pmod6$ and the rest $\equiv1\pmod6$

$\implies\prod_{i=1}^6p_i\equiv-1\pmod 6\iff(\prod_{i=1}^6p_i)+1\equiv0\pmod6$

Example$: 5,11,17$ and $7,13,19$

Answer 2

This OEIS sequence: http://oeis.org/A002110

gives the sequence of primorial numbers, which are the product of consecutive primes.

The seventh entry in the list, 30030 = 2*3*5*7*11*13.

Adding 1 to it gives 30031, which is 59*509, so it's composite, as required.

Answer 3

The simplest example I could come up with is $p_i=2$ for all $i$, so that $$p_1\cdot p_2\cdot p_3\cdot p_4\cdot p_5\cdot p_6+1=2^6+1=65=5\cdot13.$$ If you want them to be distinct, just take the first six primes: $$2\cdot3\cdot5\cdot7\cdot11\cdot13+1=59\cdot509,$$ or note that the product of any six odd primes plus one is even, hence not prime.