|Ivars Peterson's MathTrek|
September 28, 1998
The numbers 61 and 62 also happen to share a curious numerical property, says Mike Keith, a software engineer, inventor, and consultant in Salem, Ore.
Determine the prime factors of each of these numbers, then sum up the decimal digits of both the prime factors and the number itself. It turns out that 61 is a prime number, so the required sum is (6 + 1) + (6 + 1), or 14; 62 is the product of the prime numbers 31 and 2, so the sum is (3 + 1 + 2) + (6 + 2), or 14. The two sums are equal.
Other pairs of consecutive integers have the same property. Consider, for example, the pair 273 and 274: 273 = 3 x 7 x 13, and (3 + 7 + 1 + 3) + (2 + 7 + 3) = 26; 274 = 2 x 137, and (2 + 1 + 3 + 7) + (2 + 7 + 4) = 26.
Paying numerical tribute to a set of remarkable baseball feats, Keith calls such pairs Maris-McGwire-Sosa numbers. That designation also honors earlier work on the properties of numbers called Ruth-Aaron pairs, such as 714 and 715 (see "Playing with Ruth-Aaron Pairs," 6/30/97).
In a paper posted at his Web site (http://users.aol.com/s6sj7gt/maris.htm), Keith lists all the Maris-McGwire-Sosa numbers less than 1000. The first pair is 7 (7 + 7 = 14) and 8 (2 + 2 + 2 + 8 = 14). A computer search of integers up to 1 billion reveals 32,023,033 such pairs.
Keith then asks whether it is possible to find three consecutive integers that show a similar relationship. The answer is yes. Consider, for example, 212, 213, and 214: 212 = 2 x 2 x 53, and (2 + 2 + 5 + 3) + (2 + 1 + 2) = 17; 213 = 3 x 71, and (3 + 7 + 1) + (2 + 1 + 3) = 17; 214 = 2 x 107, and (2 + 1 + 0 + 7) + (2 + 1 + 4) = 17.
Keith used a computer to find sets of four, five, six, and seven consecutive integers that fit the same rule. Here's his tabulation of the set of smallest integers for which the relationship holds in each case from 2 to 7 consecutive integers.
Does it work for eight, nine, or more consecutive integers? Keith speculates that if you go high enough, you'll eventually find an example for any number of consecutive integers. "But a proof of this seems difficult," he notes.
It's also possible to study the frequency distribution of these special sets of integers. Notice that the first example of a set of seven consecutive integers starts with a number that is smaller than that for six consecutive integers. The statistical distribution pattern, however, suggests that you should not have expected to reach that initial example until you looked at numbers close to 109 . Keith's discovery of the special properties of Maris-McGwire-Sosa numbers is just one demonstration of his credo that "all numbers are interesting."
"Although I use mathematics in my job, it is of course the case that papers such as [Maris-McGwire-Sosa numbers] are a result more of my love of math than my training or the kind of math I need for my work," Keith says.
What ties it all together is Keith's use of the computer -- informed by theory -- as a tool for exploring mathematical ideas to see where they might lead.
Keith's name has come up twice before in my MathLand/MathTrek columns. I noted his unique mnemonic device encoding the first 740 decimal digits of pi in a poem modeled on Edgar Allan Poe's "The Raven" (see "A Passion for Pi," 3/11/96). I also mentioned Keith numbers in an article on particular integers that earn collective names (see "Names for Numbers," 10/27/97).
When you put words and numbers together, there's a lot out there to play with.
Copyright 1998 by Ivars Peterson
Mike Keith's paper on Maris-McGwire-Sosa numbers is available at http://users.aol.com/s6sj7gt/maris.htm. Some of Keith's other pastimes can be sampled at http://users.aol.com/s6sj7gt/mikehome.htm.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.