Triangular number AND sum of first m factorials

[Paul Dale]

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