Self-descriptive number
Encyclopedia
A self-descriptive number is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

m that in a given base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

b is b-digit
Digit
Digit may refer to:* Digit , one of several most distal parts of a limbâ€”fingers, thumbs, and toes on hands and feet* Numerical digit, as used in mathematics or computer science* Hexadecimal, representing a four-bit number...

s long in which each digit d at position n (the most significant digit being at position 0 and the least significant at position b - 1) counts how many instances of digit n are in m.

For example, in base 10, the number 6210001000 is self-descriptive because of the following reasons:

In base 10, the number has 10 digits;

It contains 6 at position 0, indicating that there are six 0s;

It contains 2 at position 1, indicating that there are two 1s;

It contains 1 at position 2, indicating that there is one 2;

It contains 0 at position 3, indicating that there is no 3;

It contains 0 at position 4, indicating that there is no 4;

It contains 0 at position 5, indicating that there is no 5;

It contains 1 at position 6, indicating that there is one 6;

It contains 0 at position 7, indicating that there is no 7;

It contains 0 at position 8, indicating that there is no 8;

It contains 0 at position 9, indicating that there is no 9.

There are no self-descriptive numbers in bases 2, 3 or 6. In bases 7 and above, there is, if nothing else, a self-descriptive number of the form , which has b - 4 instances of the digit 0, two instances of the digit 1, one instance of the digit 2, one instance of digit b - 4, and no instances of any other digits. The following table lists some self-descriptive numbers in a few selected bases:
Base Self-descriptive numbers Values in base 10
4 1210, 2020 100
100 (number)
100 is the natural number following 99 and preceding 101.-In mathematics:One hundred is the square of 10...

, 136
136 (number)
136 is the natural number following 135 and preceding 137.-In mathematics:136 is itself a factor of the Eddington number...

5 21200 1425
7 3211000 389305
8 42101000 8946176
9 521001000 225331713
10 6210001000 6210001000
16 C210000000001000 13983676842985394176
36 W21000 ... 0001000
(Ellipsis
Ellipsis
Ellipsis is a series of marks that usually indicate an intentional omission of a word, sentence or whole section from the original text being quoted. An ellipsis can also be used to indicate an unfinished thought or, at the end of a sentence, a trailing off into silence...

omits 23 zeroes)
Approx. 2.14349 × 1053

From the numbers listed in the table, it would seem that all self-descriptive numbers have digit sums equal to their base, and that they're multiples of that base. The first fact follows trivially from the fact that the digit sum equals the total number of digits, which is equal to the base, from the definition of self-descriptive number.

That a self-descriptive number in base b must be a multiple of that base (or equivalently, that the last digit of the self-descriptive number must be 0) can be proven ad absurda as follows: assume that there is in fact a self-descriptive number m in base b that is b-digits long but not a multiple of b. The digit at position b - 1 must be at least 1, meaning that there is at least one instance of the digit b - 1 in m. At whatever position x that digit b - 1 falls, there must be at least b - 1 instances of digit x in m. Therefore, we have at least one instance of the digit 1, and b - 1 instances of x. If x > 1, then m has more than b digits, leading to a contradiction of our initial statement. And if x = 0 or 1, that also leads to a contradiction.

The concept of self-descriptive numbers is similar to that of autobiographical numbers or curious numbers, except that there is no digit length requirement for autobiographical numbers. Self-descriptive numbers are like self number
Self number
A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is,...

s only in that they're both base-dependent concepts.
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