Question

One is added to the product of four consecutive positive integers. Prove that the result is never a prime
number.

Answer 1

\[
1+(n)(n+1)(n+2)(n+3)
\]

Answer 2

I would try expanding it.

Answer 3

from plugging in the first few numbers (1, 2, 3, etc), it seems that its always a square. So it should be possible to factor the expression\[n(n+1)(n+2)(n+3)+1\]as a square.

Answer 4

yeah:\[n(n+1)(n+2)(n+3)+1=(n^3+3n^2+2n)(n+3)+1\]\[((n^2(n+3)+2n)(n+3)+1=(n(n+3))^2+2n(n+3)+1=(n(n+3)+1)^2\]