Recreational mathematics offers 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 numbers that are perfect, amicable, lucky, narcissistic, weird, and so on. Many of these names also commemorate people: Mersenne, Fermat, Fibonacci, Keith, Niven, Carmichael, Stirling, Catalan, Ruth-Aaron, Rhonda. And 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 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 666, for instance, has the prime factorization 2 x 3 x 3 x 37, and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7 = 18. About 3,300 Smith numbers lie between zero and 100,000, and slightly fewer fall between 100,000 and 200,000. There are 29,928 Smith numbers below 1 million.
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 3,304p is a Smith number. However, no one has yet found a general-purpose formula for generating every possible Smith number.
In 1987, Wayne McDaniel of the University of Missouri at St. Louis proved that 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). The smallest Smith triple happens to be 73,615, 73,616, and 73,617.
Then, of course, there are Rhonda numbers, named by Kevin Brown (see http://www.mathpages.com/home/kmath007.htm) in honor of an acquaintance whose address included the number 25662. The prime factors of 25,662 are 2, 3, 7, 13, and 47. The sum of the prime factors is 2 + 3 + 7 + 13 + 47 = 72. The product of the decimal digits of the number is 2 x 5 x 6 x 6 x 2 = 720, or 10 times the sum of the prime factors.
So, a Rhonda number in base b is a number such that the product of the digits in base b equals b times the sum of its prime factors (b itself must be composite).
The first few Rhonda numbers in base 10 are 1568, 2835, 4752, 5265, and 5439.
The smallest Rhonda number is 560, which is Rhonda to base 12. There also exist numbers that are Rhonda to more than one base. The smallest of these is 1000, which is Rhonda to bases 16 and 36.
You can also prove that there are infinitely many Rhonda numbers.
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. “They are instead observing mere accidents of their representation in an arbitrary system.”
On the other hand, 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.
In the meantime, Tanya Khovanova of Princeton University gives you another way to explore the wonders of numbers. At her “Number Gossip” page at http://www.tanyakhovanova.com/Numbers/index.html, you can enter any positive integer less than 10,000 and learn its “special” properties.
For example, entering 720, you find out that it is even, composite, abundant, apocalyptic (power), evil, practical, and Ulam. Moreover, 720 is the largest factorial to contain all different digits, and 720 is the smallest number with 30 divisors.
By the way, a number is evil if it has an even number of 1s in its binary representation. Khovanova provides definitions of more than 40 named numbers.
Another source you can check is Erich Friedman’s page “What’s Special About This Number?” at http://www.stetson.edu/~efriedma/numbers.html.
When it comes to numbers, there’s always more. The fun never ends!
Originally posted: Oct. 25, 1997
Updated: July 29, 2006