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 Summing consecutive factorials (Posted on 2018-06-26)
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 No Rating

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 re: If you don't have to start with 1 ... | Comment 3 of 7 |

Wolfram lists some sums (not necessarily consecutive) here

The only consecutive sum not previously listed by the first two posters is
0! + 1! + 2! = 4

The Full Wolfram list, containing all sums < 10^12 is

0!+1!+2! = 2^2
1!+2!+3! = 3^2
1!+2!+3!+6! = 27^2
1!+2!+3!+6!+7!+8!+10! = 1917^2
1!+2!+3!+6!+9! = 603^2
1!+2!+3!+7!+8! = 213^2
1!+4! = 5^2
1!+4!+5!+6!+7!+8! = 215^2
1!+4!+8!+9! = 635^2
1!+5! = 11^2
1!+5!+6! = 29^2
1!+7! = 71^2
4!+5! = 12^2
4!+5!+7! = 72^2
1!+2!+3!+7!+8!+9!+10!+11!+12!+13!+14!+15!=1183893^2

 Posted by Steve Herman on 2018-06-26 10:19:37

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