How is this proof finished? I was told it is proven

[AcousticBruce]

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

So let n be a natural number. So f(n) = n(n+1)(n+2)(n+3)

n --- 1 --- 2 --- 3 --- 4 --- 5 --- 10
f(n)-24--120-360-840-1080-17160

The conjecture I want to prove is F(n) + 1 is always a square.

n(n+1)(n+2)(n+3) = (n²+3n)(n²+3n+1)

so

[(n²+3n-1)+1)][(n²+3n+1)-1]+1

and because a²-b² = (a+b)(a-b)

[(n²+3n+1)²-1)+1] = (n²+3n)²

So this proves that f(n) + 1 is in fact a square.

My question is how does this prove that f(n) is NOT a square? I mean it seems obvious, but I am trying to learn to prove things.

 

[TeethWhitener]

Think about the size of the gap between consecutive square numbers.

 

[AcousticBruce]

This is exactly what a friends said. The size gap. Ill think on this the rest of the day and see if I can find why that matters. Thanks.

 

[TeethWhitener]

Just try a test case: what's the gap between x² and (x-1)²?

 

[HomogenousCow]

By the way, n(n+1)(n+2)(n+3) = (n2+3n)(n2+3n+2), so f(n)+1 = (n^2+3n+1)^2
The argument is correct, however.