|Ivars Peterson's MathTrek|
September 8, 1997
Note: To avoid confusion with a widely used primary-school curriculum known as MathLand, I have changed the name of my weekly online math column to Ivars Peterson's MathTrek, starting with this article.
Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.
To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor. The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.
Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.
Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.
This particular sieving process yields certain numbers that permanently escape getting killed. That's why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.
1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195
What's remarkable is that the "luckies," though generated by a sieve based entirely on a number's position in an ordered list, share many properties with primes. For example, there are 25 primes less than 100, and 23 luckies less than 100. Indeed, it turns out that primes and luckies come up about equally often within given ranges of integers. The distances between successive primes and the distances between successive luckies also keep increasing as the numbers increase. In addition, the number of twin primes -- primes that differ by 2 -- is close to the number of twin luckies.
Perhaps the most famous problem involving primes still unsolved is the Goldbach conjecture, which states that every even number greater than 2 is the sum of two primes. Luckies are featured in a similar conjecture, also unsolved: Every even number is the sum of two luckies. Computer searches have reached at least 100,000 without finding an exception.
Martin Gardner describes many more features of lucky numbers in a delightful article in a recent issue of The Mathematical Intelligencer. "There is a classic proof by Euclid that there is an infinity of primes," he writes. "Although it is easy to show there is also an infinity of lucky numbers, the question of whether an infinite number of luckies are primes remains, as far as I know, unproved."
How did the topic of lucky numbers happen to come up? The house where Gardner grew up in Tulsa, Okla., had the address 2187 S. Owasso. "Of course I never forgot this number," he says. It also happens to be one of the lucky numbers.
Gardner's imaginary friend, the noted numerologist Dr. Irving Joshua Matrix, can readily find additional remarkable properties associated with that number. Exchange the last two digits of 2187 to make 2178, multiply by 4, and you get 8712, the second number backward. Take 2187 from 9999 and the result is 7812, the number in reverse. Moreover, the first four digits of the constant e, 2718, and the number of cubic inches in a cubic foot, 12^3 = 1728, are each permutations of 2187!
However, to those inclined to seeing meaning in certain numbers, Dr. Matrix issues the following warning: "Every number has endless unusual properties."
Recently, I had the great pleasure of meeting Martin Gardner for the first time. After the Atlanta Mathfest last month, my family and I happened to be vacationing near Hendersonville, N.C., where he lives, and we took advantage of that amazing coincidence to pay him a visit.
I have long been a great fan of Martin Gardner's writings, starting with his "Mathematical Games" columns in Scientific American. I obtained my first issue of Scientific American in 1962, when I was in the ninth grade. It featured a series of articles on Antarctica, which was the topic of study in my geography class. But it was Gardner's discussion of tests that show whether a large number can be divided by a number from 2 to 12 that really caught my attention. From that time, I never missed his articles.
Martin Gardner's last Scientific American column appeared in May 1986. The final collection of his columns to be published in book form is now available in The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications.
I, for one, am very glad that Martin Gardner continues to write about the joys of mathematics. Finding one of his articles, whether in The Mathematical Intelligencer, Math Horizons, or elsewhere, is always a treat.
Copyright © 1997 by Ivars Peterson.
______. 1997. Lucky numbers and 2187. The Mathematical Intelligencer 19(No. 2):26.
______. 1962. Mathematical games: Tests that show whether a large number can be divided by a number from 2 to 12. Scientific American 207(September):232.
Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.
______. 1988. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W.H. Freeman.