The number 3435 = 33+44+33+55 is well-known as the Münchhausen number in the base 10 and its length (number of digits) is 4.
More examples:
The number 96446 is the Münchhasuen number in the base 9 and its length is 6, because it has 6 digits in the base 9:
9644610 = 1562629
digits 1, 5, 6, 2, 6, 2
96446 = 11+55+66+22+66+22
The number 20017650854 is the Münchhasuen number in the base 12 and its length is 10, because it has 10 digits in the base 12:
2001765085410 = 3a67a5483212
digits 3, a, 6, 7, a, 5, 4, 8, 3, 2
20017650854 = 33+1010+66+77+1010+55+44+88+33+22 (the digit a is ten)
Bases can be considered arbitrary. Then it is not difficult to prove that there are infinitely many Münchhasuen numbers of length 2 in some base. In particular, 1+nn is of length 2 in the base 1+nn-n for each n greater than 1.
But what about length 3? Are there infinitely many Münchhausen numbers of length 3? Up to now, we know only these Münchhasuen numbers of length 3:
29 in base 4
55 in base 4
3153 in base 25
49782 in base 91
46661 in base 215
New: 823545 in base 904 (June 21, 2020; by mersenneforum.org user axn)
New: 823545 in base 904 (June 21, 2020; by mersenneforum.org user axn)
