Exercise 1121Number theory › PrimesOlympiadAdded 3 Nov 2026

Consecutive primes and products

#consecutive primes#product#gap

Statement

Show that for consecutive primes $p_n<p_{n+1}$ , the product $p_np_{n+1}$ is not divisible by any prime smaller than $p_n$ except $p_n$ itself.

Hint

The prime divisors of $p_np_{n+1}$ are exactly $p_n$ and $p_{n+1}$ .

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Prime divisors. $p_n$ and $p_{n+1}$ are the only prime divisors of $p_np_{n+1}$ .

Consequence. Any prime divisor of the product is $\ge p_n$ .

Conclusion. No prime $<p_n$ divides $p_np_{n+1}$ .