Given the list of n primes puzzle

n primes puzzle

Given the list of the first n primes, 2, 3, 5,…, the product of the primes plus one will be coprime to the list of primes used. For example:

$\begin{align*}E_1&=2+1\\&=3\end{align*}$

and 3 is coprime to 2.

$\begin{align*}E_2&=2\times3+1\\&=7\end{align*}$

and 7 is coprime to 2 and 3.

Will $E_n$ always be a prime number?

Will $E_n$ ever be a square number?

Reveal Solution

The first value of $E_n$ that is not a prime number is $E_6$ .

$\begin{align*}E_6&=2\times3\times5\times7\times11\times13+1\\&=30031\\&=59\times509\end{align*}$

$E_n$ will never be a square number.

$E_n=p_1\times p_2 \times \dots \times p_n +1$

If $E_n$ is a square number this can be written as:

$\begin{align*}m^2 &= p_1 \times p_2 \times \dots\times p_n +1\\p_1 \times p_2 \times \dots\times p_n &= m^2 -1\\&=(m+1)(m-1)\end{align*}$

$(m+1)(m-1)$ must either be both odd or both even for this result to hold.

$p_1 \times p_2 \times \dots\times p_n$ has a single even factor of 2. $(m+1)$ and $(m-1)$ cannot both be odd as this would result in $p_1 \times p_2 \times \dots\times p_n$ being odd. $(m+1)$ and $(m-1)$ cannot both be even as this would mean 4 was a factor of $p_1 \times p_2 \times \dots\times p_n$ .