4 Consecutive Integers cannot be Square-Free

Theorem

Let $n, n + 1, n + 2, n + 3$ be four consecutive positive integers.

At least one of these is not square-free.

Proof

Exactly one of $n, n + 1, n + 2, n + 3$ is divisible by $4 = 2^2$.

Thus, by definition, one of these is not square-free.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $29$