MathLabs

Problem 5

Prove that for every positive integer nn there exist nn consecutive positive integers none of which is a prime or a power of a prime.
Step 1 of 4: Choose the factorial square
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In plain words

Factorials make every small offset divide the large base term.

N=n+1,Xr=(N!)2+r(2≤r≤N)N=n+1,\qquad X_r=(N!)^2+r\quad(2\le r\le N)
Detailed analysis

Set N=n+1N=n+1 and define Xr=(N!)2+rX_r=(N!)^2+r for r=2,3,…,Nr=2,3,\ldots,N. These are consecutive integers, from (N!)2+2(N!)^2+2 through (N!)2+N(N!)^2+N, so there are exactly N−1=nN-1=n of them.