Discover Something
Let N be one more than the product of four consecutive positive integers. What can you say about N? Prove it.
(Of course, what you say about N must have some substance. Trivial statements are unacceptable.)
Please submit your work to mathpotw@acu.edu by next Thursday (2/13/2020).
Solution to Discover Something
A partially correct solution was submitted by: Wyatt Witemeyer.
A little bit of calculation leads one to conjecture that the product of four consecutive integers plus one is a perfect square. We could even toss in the fact that the square is odd, since the product of four consecutive integers is even, and an even plus one is odd.
So let’s prove that n(n+1)(n+2)(n+3) + 1 is a perfect square for any integer n. There are many approaches to proving this result, but the following one seems rather elegant. Regrouping leads to
n(n+3)(n+1)(n+2) + 1 =
(n2+3n)(n2+3n + 2) + 1 =
([n2+3n+1] – 1)([n2+3n+1] + 1) + 1 =
([n2+3n+1]2 – 1) + 1 =
[n2+3n+1]2.
