Talk:Factorions

Limit of 1499999 in base 10

The following was removed from the wikipedia page in September 2019, on the grounds "described above for arbitrary base b".
Since I rather strongly disagree on that last point, I have replicated the original/deleted text here:

Upper bound (in base 10)

If n is a natural number of d digits that is a factorion, then 10^{d − 1} ≤ n ≤ 9!d. This fails to hold for d ≥ 8 thus n has at most seven digits, and the first upper bound is 9,999,999. But the maximum sum of factorials of digits for a seven-digit number is 9!*7 = 2,540,160 establishing the second upper bound. Going further, since no number bigger than 2540160 is possible, the first digit of a seven-digit number can be at most 2. Thus, only six positions can range up until 9 and 2!+6*9!= 2177282 becomes a third upper bound. This implies, if n is a seven-digit number, either the second digit is 0 or 1 or the first digit is 1. If the first digit is 2 and thus the second digit is 0 or 1, the numbers are limited by 2!+1!+5*9! = 1814403 - a contradiction to the first digit being 2. Thus, a seven-digit number can be at most 1999999, establishing our fourth upper bound.

All factorials of digits at least 5 have the factors 5 and 2 and thus end on 0. Let 1abcdef denote our seven-digit number. If all digits a-f are all at least 5, the sum of the factorials - which is supposed to be equal to 1abcdef - will end on 1 (coming from the 1! in the beginning). This is a contradiction to the assumption that f is at least 5. Thus, at least one of the digits a-f can be at most 4, which establishes 1!+4!+5*9!=1814425 as fifth upper bound. Assuming n is a seven-digit number, the second digit is at most 8. There are two cases: If a is at least 5, by the same argument as above one of the remaining digits b-f has to be at most 4. This implies an upper bound (since a is at most 8) of 1!+8!+4!+4*9!= 1491865, a contradiction to a being at least 5. Thus, a is at most 4 and the sixth upper bound is 1499999.

A computer can check all numbers from 40585 to 1499999, verifying that 40585 is the largest factorion in base 10.

Quite incorrectly, almost all entries (bar Pascal, Perl[2], Phix[2], Sidef) also use that limit for bases 9, 11, and 12... --Pete Lomax (talk) 22:42, 15 January 2022 (UTC)