1!=1^2; 1!+2!+3!=3^2
Are there any other sums of n consecutive factorials that sum up to a perfect square?
Either list them all or prove there are none.
case 1: n<5
1!=1=1^2
1!+2!=3
1!+2!+3!=9=3^2
1!+2!+3!+4!=33
so the only solutions here are n=1 and 2 as given in the description.
case 2: n>=5
for n>=5 n! is divisible by 10
1!+2!+3!+4!=33=3 mod 10
thus the sum for n>=5 is congruent to 3 mod 10
however, the prefect squares mod 10 are restricted to [0,1,4,5,6,9].
Thus there none of these can be a perfect square.
Thus 1!=1^2 and 1!+2!=3^2 are the only such occurances.

Posted by Daniel
on 20180626 10:02:58 