Question

Prove or disprove that${\mathit{p}}_{\mathbf{1}}{\mathit{p}}_{\mathbf{2}}\mathbf{\dots }{\mathit{p}}_{\mathbf{n}}\mathbf{+}\mathbf{1}$is prime for every positive integer n, where role="math" localid="1668525741485" ${\mathit{p}}_{\mathbf{1}}\mathbf{,}{\mathit{p}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{p}}_{\mathbf{n}}$are the n smallest prime numbers.

Short answer

Disproven.

Step-by-step solution

Step 1

An integer p larger than x is called prime if the only positive factors of p are 1 and p.

The smallest prime numbers are

$2,3,5,7,11,13,17,19,\dots$

The claim is that $p_1{p}_2\dots p_n+1$is prime.

$2+1=3$is prime

$2\cdot 3+1=7$is prime

$2\cdot 3\cdot 5+1=31$is prime

$2\cdot 3\cdot 5\cdot 7+1=211$is prime

$2\cdot 3\cdot 5\cdot 7\cdot 11+1=2311$is prime

$2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13+1=30031$is not prime, because $59\cdot 509=30031$.

Thus, the given statement is false when the first $n=6$prime numbers are used.