Exercise 541Number theory › PrimesHigh schoolAdded 3 Nov 2026

Consecutive primes and coprimality

#consecutive primes#coprime#primorial

Statement

Show that the product of the first $n$ primes plus $1$ is coprime to each of these primes.

Hint

If $p_i$ divides $p_1\cdots p_n+1$ , then $p_i$ divides $1$ .

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Setup. Let $N=p_1\cdots p_n+1$ .

Coprimality. For any $p_i$ : $p_i\mid p_1\cdots p_n$ and if $p_i\mid N$ , then $p_i\mid N-p_1\cdots p_n=1$ , contradiction.

Conclusion. $\gcd(N,p_i)=1$ for all $i$ . So any prime divisor of $N$ is larger than $p_n$ .

Application. This gives another proof that there are infinitely many primes.