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# prove that1! + 2! + 3! + .... + n! cant be a pefect power for any n>3

Swapnil Saxena
102 Points
9 years ago

The question can be solved by the principle of cyclicity of unit place.

The question simply wanna say that root(1!+2!+3!+4!+5!....) cant ba an integer fr n>3. It is irrational

Now for n>3, observe that The unit place of summation is always 3 (Simply the 5!,6! have their unit place as 0 adding 33(1!+2!+3!+4!) make no difference n unit place so this is logically explained)

As we know that the last digit of the nth power can be easily predicted by using the rule that the last digit repeats after a interval of 4.

Lets make a list of the unit place of unit place of n the power of a no

Ex:

2,4,8,16,32,.. have their last digits 2,4,8,6,2,4 ...

3,9,27,81,243 have their last digits as 3,9,7,1,3...

Similarly make a list frm 4 to 9.

You will observe that no no. has its square power as 3. So simply the summation cant be a perfect square.  So root (1!+2!+3!+...)

Hence proved.

Sathya
35 Points
9 years ago

Can u prove it with modular algebra?