# Perfect digit-to-digit invariant

A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number) is a natural number that is equal to the sum of its digits each raised to a power equal to the digit. 0 and 1 are PDDIs in any base (using the convention that 00 = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 (sequence A046253 in the OEIS). Note that the second of these is only a PDDI under the convention that 00 = 0, but this is standard usage in this area.   More generally, there are finitely many PDDIs in any base. This can be proved as follows:

Let be a base. Every PDDI in base is equal to the sum of its digits each raised to a power equal to the digit. This sum is less than or equal to , where is the number of digits in , because is the largest possible digit in base . Thus, The expression increases linearly with respect to , whereas the expression increases exponentially with respect to . So there is some such that There are finitely many natural numbers with fewer than k digits, so there are finitely many natural numbers satisfying the first inequality. Thus, there are only finitely many PDDIs in base .

In all bases 1 is a PDDI.
In base 3 there are 2 PDDI's, namely 12 and 22. (5 and 8 in decimals)
In base 4 there are 2 PDDI's, namely 131 and 313. (29 and 55 in decimals)
In base 6 there are 2 PDDI's, namely 22352 and 23452. (3164 and 3416 in decimals)
In base 7 there is 1 PDDI's, namely 13454. (3665 in decimals)
In base 9 there are 3 PDDI's, namely 31, 156262 and 1656547. (28, 96446 and 923362 in decimals)

When the convention is used the following numbers are also PDDI's

In all bases 0 is a PDDI.
In base 4 there is one additional PDDI, namely 130. (28 in decimal)
In base 5 there are 2 PDDI's, namely 103 and 2024. (28 and 264 in decimal)
In base 8 there are 2 PDDI's, namely 400 and 401. (256 and 257 in decimal)
In base 9 there are 3 additional PDDI's, namely 30, 1647063 and 34664084. (27, 917139 and 16871323 in decimal)

## References

1. van Berkel, Daan (2009). "On a curious property of 3435". arXiv:0911.3038 [math.HO].
2. Narcisstic Number, Harvey Heinz
3. Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. p. 185. ISBN 0-14-026149-4.