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.

Therefore E_n will never be a square number.

Problem Page Coordinator: Stephen Lee CMath MIMA – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2015)

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