Product of 4 Consecutive Integers is One Less than Square/Lemma

Lemma for Product of 4 Consecutive Integers is One Less than Square

Let $a$, $b$, $c$ and $d$ be consecutive integers.

Let us wish to prove that the product of $a$, $b$, $c$ and $d$ is one less than a square.

Then it is sufficient to consider $a$, $b$, $c$ and $d$ all strictly positive.

Proof

If $0 \in \set {a, b, c, d}$ then $a b c d = 0$ which is $1$ less than $1^2$.

If $a$, $b$, $c$ and $d$ are all negative, then:

$a b c d = \paren {-a} \paren {-b} \paren {-c} \paren {-d}$

Hence it is sufficient to consider $a$, $b$, $c$ and $d$ all strictly positive.

$\blacksquare$