Perfect digit-to-digit invariant
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 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 base 2 the only PDDI is 1.
In base 3 there are 3 PDDIs, namely 1, 12 and 22. (1, 5, 8 in decimals)
In base 4 there are also 3 PDDIs, namely 1, 131 and 313. (1, 29, 55 in decimals)
In base 5 there are none except for the trivial case 1.
In base 6 there are 3 PDDIs, namely 1, 22352 and 23452. (1, 3164, 3416 in decimals)
In base 7 there are 2 PDDIs, namely 1 and 13454. (1, 3665 in decimals)
In base 8 there is again only the trivial case 1.
In base 9 there are 4 PDDIs, namely 1, 31, 156262 and 1656547. (1, 28, 96446, 923362 in decimals)