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 3 of 4: Use a prime divisor of the offset

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In plain words

The added $1$ prevents the chosen prime from accounting for the whole factorization.

p_r\mid r,\quad p_r\mid X_r,\quad p_r\nmid\frac{X_r}{r}

Detailed analysis

Choose a prime $p_r$ dividing $r$ . Since $N!/r$ is an integer, the quotient $N!(N!/r)$ is divisible by $r$ , so $N!(N!/r)+1$ is congruent to $1$ modulo $r$ , and in particular is not divisible by $p_r$ . Thus $X_r$ is divisible by $p_r$ but its quotient by $p_r$ is still greater than $1$ .