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Summing consecutive factorials (Posted on 2018-06-26) Difficulty: 1 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
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analytical solution | Comment 1 of 7
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 2018-06-26 10:02:58
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