Why do Smith numbers have to be composite numbers?

Question

As you may know, a Smith number is a number that if all the digits are added together that answer is equal to the sum of its prime factors' digits.

Why are 2 and 3 not Smith numbers?

Answer 1

Smith numbers appeared in 1982 when Albert Wilansky "published" an "article" in the The Two-Year College Mathematics Journal. The article is really short and just gives the definition and an example and some questions. According to the definition a Smith number is a composite number. Wilansky came across the numbers because he noticed his brother-in-law's (named Smith) phone number $4937775$ is a Smith number.

As a mathematician Wilansky probably immediately had questions. How many Smith numbers are there? Specifically he wondered in the "article" if the phone number is the largest Smith number. He then probably thought about it and realized that interesting problems come from these numbers. Now, if your definition says that prime numbers are Smith numbers, then you have already answered the question about how many there are. So instead of saying "Smith numbers that are not prime" every time, he simply required the number to be composite in the definition.

If you want to read more about this, you can take a look at a couple of articles (if you are at a university, you should be able to find these)

• Smith Numbers, Underwood Dudley, Mathematics Magazine, Vol. 67, No. 1 (Feb., 1994), pp. 62-65
• The Sum of Digits Function and its Application to A Generalization of the Smith Number Problem, Wayne McDaniel, Samuel Yates, Nieuw Archief Voor Wiskunde, 7, No, 1-2, March/July, (1989), 39-51.

Answer 2

All prime numbers fulfill the sum-of-digits-condition, and are therefore relatively uninteresting in this context. Hence, the Smith numbers are specified to be non-prime.

Another way of looking at it is: All prime numbers fulfill this condition. What other numbers do so? The Smith numbers is the answer.