3!+3=9=32
4!+1=25=52
5!+1=121=112
6!+9=729=272
7!+1=5041=712
8!+81=40401=2012
In the above examples, the number that must be added to each factorial to reach the first perfect square beyond it is itself a perfect square. (With the exception of 2! and 3!)
Make a conjecture about whether the observed pattern continues before you try to prove it or find a counterexample.
A Counterexample
