Triangular number AND sum of first m factorials
01-10-2018, 06:35 AM (This post was last modified: 01-10-2018 10:59 AM by Paul Dale.)
Post: #9
 Paul Dale Senior Member Posts: 1,757 Joined: Dec 2013
RE: Triangular number AND sum of first m factorials
I've got a proof that there are only three such numbers.

Consider the last pair of digits in $$\sum_1^n i!$$, from n=9 onwards these never change because subsequent factorial terms will always have a factor of 100 present. These digits are '13'.

Note that n is triangular iff 8n+1 is a perfect square. For the sum of factorials to be triangular, the last two digits must therefore be '05'. Checking all possibilities shows that there are no square numbers that end '05'.

Thus, numbers of the desired form must have n < 9. Checking all cases reveals that only 1, 3 and 153 have the desired properties.

Pauli
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 Messages In This Thread Triangular number AND sum of first m factorials - Joe Horn - 01-09-2018, 04:31 PM RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-09-2018, 08:53 PM RE: Triangular number AND sum of first m factorials - Dieter - 01-09-2018, 10:19 PM RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-09-2018, 11:00 PM RE: Triangular number AND sum of first m factorials - Valentin Albillo - 01-09-2018, 10:16 PM RE: Triangular number AND sum of first m factorials - John Keith - 01-09-2018, 11:10 PM RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-10-2018, 04:03 AM RE: Triangular number AND sum of first m factorials - Joe Horn - 01-10-2018, 04:58 AM RE: Triangular number AND sum of first m factorials - Paul Dale - 01-10-2018 06:35 AM RE: Triangular number AND sum of first m factorials - Joe Horn - 01-11-2018, 03:01 AM RE: Triangular number AND sum of first m factorials - Paul Dale - 01-11-2018, 10:21 AM RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-11-2018, 06:29 PM RE: Triangular number AND sum of first m factorials - John Keith - 01-11-2018, 10:43 PM RE: Triangular number AND sum of first m factorials - John Keith - 01-11-2018, 10:30 PM RE: Triangular number AND sum of first m factorials - John Cadick - 01-11-2018, 02:22 PM

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