|Ivars Peterson's MathTrek|
October 27, 1997
Recreational mathematics furnishes a vast playing field for amateur and professional mathematicians alike. It combines a sense of play with the joy of discovery. Sometimes the results are mathematically trivial; occasionally they lead to new mathematical insights.
Whole numbers or integers are often the subject of such pursuits. Once someone discovers an interesting pattern or type of behavior, those particular numbers are likely to earn a collective name. So we have perfect numbers, amicable numbers, lucky numbers, Mersenne numbers, Fermat numbers, Fibonacci numbers, Keith numbers, Niven numbers, Carmichael numbers, Stirling numbers, Catalan numbers, Ruth-Aaron numbers, Rhonda numbers, and so on. The list keeps growing!
One such curiosity came about in 1982 as the result of a telephone call. When phoning his brother-in-law, mathematician Albert Wilansky of Lehigh University in Bethlehem, Penn., noticed that the telephone number had a striking property. The number, 493-7775 (4,937,775), is composite, meaning that it can be expressed as the product of prime numbers: 3 x 5 x 5 x 65,837. Interestingly, when the digits of the original number are added together, the result (42) equals the sum of the digits of the prime factors (3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42). This discovery marked the birth of Smith numbers, named for Wilansky's brother-in-law.
The smallest Smith number is 4 because the number's factors, 2 x 2, when added together also equal 4. The next one is 22; then comes 27. Overall, there are 376 Smith numbers among the first 10,000 positive integers. The integer 6,036, for instance, has the prime factorization 2 x 2 x 3 x 503, and 6 + 0 + 3 + 6 = 2 + 2 + 3 + 5 + 0 + 3. About 3,300 Smith numbers lie between zero and 100,000, and slightly fewer fall between 100,000 and 200,000.
Investigations by both amateur and professional mathematicians have revealed that special patterns of digits automatically produce Smith numbers. For example, if p is a prime whose digits are all 1s, then 3304p is a Smith number. However, no one has yet found a general-purpose formula for automatically generating every possible Smith number.
In 1985, Wayne McDaniel of the University of Missouri at St. Louis managed to prove there are infinitely many Smith numbers. Others later identified palindromic Smith numbers, such as 12,345,554,321, and investigated Smith brothers (consecutive Smith numbers, such as 728 and 729)Is this serious math? To some mathematicians, anything having to do with the decimal digits of a number can't be counted as worthy of attention.
Those digits are merely a consequence of having chosen base 10 for expressing the numbers. "Thus, those who investigate Smith numbers are not trying to penetrate deep into the secrets of integers," comments Underwood Dudley of DePauw University in Greencastle, Ind. "They are instead observing mere accidents of their representation in an arbitrary system."
At the same, however, such studies can sometimes prove useful in various applications of mathematics -- particularly because we use base 10 numbers extensively in our everyday lives, from measuring distances to adding up the cost of groceries bought at the supermarket.
There's also a chance that the study of numbers expressed in a certain base highlights something unusual in the realm of mathematics -- something that distinguishes numbers expressed in one base from those expressed in another. Whether there is more to Smith numbers than mere happenstance remains to be seen. Copyright1997 by Ivars Peterson.
Dudley, Underwood. 1994. Smith numbers. Mathematics Magazine 67(February):62-65.
Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Washington, D.C.: Mathematical Association of America.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.
An annotated index of websites devoted to various types of numbers can be found in the Math Forum Internet Resource Collection at http://forum.swarthmore.edu/~steve/steve/numbers.desc.html.