Definition:Smith Number
Jump to navigation
Jump to search
Definition
A Smith number is a composite number for which the sum of its digits is equal to the sum of the digits in its prime decomposition.
Sequence of Smith Numbers
The sequence of Smith numbers begins:
- $4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, \ldots$
Also known as
Smith numbers are also referred to by some sources as joke numbers.
Also see
- Definition:Smith Brothers
- Results about Smith numbers can be found here.
Historical Note
The term Smith number was coined by Albert Wilansky.
He noticed the property in the phone number ($493$ - $7775$) of his brother-in-law Harold Smith:
- $4937775 = 3 × 5 × 5 × 65837$
while:
- $4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42$
The full text of the article in which this appeared:
- A Smith number is a composite number the sum of whose digits is the sum of all the digits of its prime factors. The (rather startling) reason for the name is mentioned below.
- Examples. $9985 = 5 \times 1997$, $9 + 9 + 8 + 5 = 5 + 1 + 9 + 9 + 7$, $6036 = 2 \times 2 \times 3 \times 503$, $6 + 0 + 3 + 6 = 2 + 2 + 3 + 5 + 0 + 3$.
- The number of Smith numbers between $n$ thousand and $n$ thousand $+ 999$ for $n = 0, 1, 2, \ldots, 9$, is, respectively, $47$, $32$, $42$, $28$, $33$, $32$, $32$, $37$, $37$, $40$.
- I wonder whether there are infinitely many Smith numbers.
- The largest Smith number known is due to my brother-in-law H. Smith who is not a mathematician. It is his telephone number: $4937775$!
Sources
- 1982: A. Wilansky: Smith Numbers (Two-Year College Math. J. Vol. 13: p. 21) www.jstor.org/stable/3026531
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4,937,775$
- Feb. 1994: Underwood Dudley: Smith Numbers (Math. Mag. Vol. 67, no. 1: pp. 62 – 65) www.jstor.org/stable/2690561
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4,937,775$
- Weisstein, Eric W. "Smith Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmithNumber.html