Smith numbers first part

Smith numbers are a class of positive integers that have an interesting property related to the sum of their digits. They were first studied by the American mathematician Albert Wilansky in 1955.

Definition of Smith numbers

A Smith number is a positive integer that has the property that the sum of its digits is equal to the sum of the digits of its prime factors.

Examples of Smith numbers

Some examples of Smith numbers are:

- 4 (2 × 2, sum of digits: 4)
- 6 (2 × 3, sum of digits: 6)
- 9 (3 × 3, sum of digits: 9)
- 22 (2 × 11, sum of digits: 2 + 2 = 4, sum of digits of prime factors: 2 + 1 + 1 = 4)
- 27 (3 × 3 × 3, sum of digits: 2 + 7 = 9, sum of digits of prime factors: 3 + 3 + 3 = 9)

Properties of Smith numbers

Smith numbers have some interesting properties:

- All Smith numbers are composite numbers (that is, they are not prime).
- The sum of the digits of a Smith number is always equal to the sum of the digits of its prime factors.
- Smith numbers can have any number of digits.

Conjectures and results about Smith numbers

Although Smith's numbers have been studied for several decades, there are still many conjectures and open results about them. Some of the most interesting conjectures are:

- The conjecture that all Smith numbers have a finite number of prime factors.
- The conjecture that the sum of the digits of a Smith number is always an even number.

In summary, Smith numbers are a class of positive integers that have an interesting property related to the sum of their digits and the sum of the digits of their prime factors. Although they have been studied for several decades, there are still many conjectures and open results about them.