Problem 5

Prove that for every positive integer

n

there exist

n

consecutive positive integers none of which is a prime or a power of a prime.

Step 4 of 4: Exclude primes and prime powers

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In plain words

A prime power has only one prime in its factorization; our construction forces a second one.

X_r\text{ has at least two distinct prime divisors}

Detailed analysis

The number $X_r$ has a prime divisor $p_r$ and also a remaining factor greater than $1$ not divisible by $p_r$ . Therefore it has at least two distinct prime divisors. It is neither a prime nor a power of one prime. The $n$ consecutive numbers $X_2,\ldots,X_N$ prove the claim.