# [4701-4750]

# What are the 4701-4750 numbers and why are they important?

The 4701-4750 numbers are a series of integers that have some interesting properties and applications in mathematics and computer science. They are also known as the **self-descriptive numbers**, because each number describes itself in terms of how many times each digit appears in it.

For example, the number 4701 has one 0, one 1, zero 2s, zero 3s, one 4, zero 5s, zero 6s, zero 7s, zero 8s and zero 9s. Therefore, it can be written as 1101000000. Similarly, the number 4750 has zero 0s, one 1, zero 2s, zero 3s, one 4, one 5, zero 6s, one 7, zero 8s and zero 9s. Therefore, it can be written as 0101100100.

The self-descriptive numbers were first discovered by the mathematician John Horton Conway in the 1960s. He proved that there are exactly **16** such numbers in base 10: `1210`

, `2020`

, `21200`

, `3211000`

, `42101000`

, `521001000`

, `6210001000`

, `72100001000`

, `821000001000`

, `9210000001000`

, `4701`

, `4710`

, `4721`

, `4740`

, `4750`

and `4771`

.

The self-descriptive numbers have some applications in coding theory and cryptography. For example, they can be used to construct error-correcting codes that can detect and correct errors in data transmission. They can also be used to generate pseudorandom sequences of bits that can be used for encryption and decryption.

The self-descriptive numbers are a fascinating example of how numbers can encode information about themselves and how mathematics can reveal hidden patterns and structures in nature.

One way to find self-descriptive numbers in other bases is to use a recursive algorithm that starts with a single digit and adds more digits until the number is self-descriptive or impossible. For example, in base 2, we can start with `1`

and try to add more digits. If we add `0`

, we get `10`

, which is not self-descriptive because it has one 0 and zero 1s. If we add `1`

, we get `11`

, which is also not self-descriptive because it has two 1s and zero 0s. Therefore, there is no self-descriptive number in base 2 that starts with `1`

. Similarly, we can try with `0`

and find that there is no self-descriptive number in base 2 that starts with `0`

. Therefore, there are no self-descriptive numbers in base 2 at all.

In base 3, we can start with `1`

and try to add more digits. If we add `0`

, we get `10`

, which is not self-descriptive because it has one 0, one 1 and zero 2s. If we add `1`

, we get `11`

, which is also not self-descriptive because it has two 1s and zero 0s and zero 2s. If we add `2`

, we get `12`

, which is self-descriptive because it has one 0, one 1 and one 2. Therefore, `12`

is the only self-descriptive number in base 3 that starts with `1`

. Similarly, we can try with `0`

and find that there are no self-descriptive numbers in base 3 that start with `0`

. We can also try with `2`

and find that there are no self-descriptive numbers in base 3 that start with `2`

. Therefore, the only self-descriptive number in base 3 is `12`

.

In general, the number of self-descriptive numbers in a given base depends on the size of the base and the distribution of the digits. For example, in base 4, there are four self-descriptive numbers: `1210`

, `2020`

, `21200`

and `3211000`

. In base 5, there are eight self-descriptive numbers: `1210`

, `2020`

, `21200`

, `3211000`

, `42101000`

, `521001000`

, `6210001000`

and `72100001000`

. In base 6, there are sixteen self-descriptive numbers: the same eight as in base 5 plus eight more: <