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 2 of 4: Factor each term by its offset

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In plain words

Every constructed number has the offset as a nontrivial factor.

r\mid X_r,\qquad X_r=r\left(N!\frac{N!}{r}+1\right)

Detailed analysis

Because $r\le N$ , we have $r\mid N!$ , hence $r\mid(N!)^2$ and $r\mid X_r$ . More explicitly, $X_r=r\left(N!\frac{N!}{r}+1\right)$ . The second factor is an integer greater than $1$ .