# Prove an expression is not a square

1. Nov 16, 2008

### hew

1. The problem statement, all variables and given/known data
prove that 1!+2!+...+n! is not a square if n>=4

2. Relevant equations

3. The attempt at a solution
in the previous part I had to prove that a sqare cannot end in 2,3,7,8 and did this by working in mod10. Do you use this to prove the question? I have written n as 10q+r where q,r are rational numbers and r=0,1,2,...,9 but I dont know where to go from here.
I would prefer a hint rather than the full solution. Thankyou.

2. Nov 16, 2008

### hew

I think i may have solved it. But do tell me if its wrong.
5!=5*4*3*2*1 and is therefore congruent to 0 mod 10 because of the 5 and 2.
so that means that all of the other factorials greater than 5 will also be congruent 0 mod 10. Therefore the remainder will come from adding 1!+2!+3!+4! which equals 32. as this is congruent 2 mod 10 all expressions for n>4 will also be conguent 2 mod 10...meaning that they cannot be squares as the earlier proof showed that numbers ending with 2 mod 10 cannot be squares.

3. Nov 17, 2008