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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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re(2): If you don't have to start with 1 ... | Comment 4 of 7 |
(In reply to re: If you don't have to start with 1 ... by Steve Herman)

good point, didn't think about changing the starting point.  Change my analysis to base 7, it can be seen that there are no further examples starting at zero.

for n<7 the only example is 0!+1!+2!=4=2^2

if n>=7 then 0!+1!+..+6!=4 mod 7 thus the sum is 4 mod 7 for all n>=7.  However there are no prefect squares 4 mod 7.  Thus there are no further examples for n>=7 starting at zero.

  Posted by Daniel on 2018-06-26 12:00:27
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