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."
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 Goldbach's 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 so far not found an exception.
Martin Gardner described many more features of lucky numbers in a delightful article, "Lucky numbers and 2187," in a 1997 issue of The Mathematical Intelligencer. "There is a classic proof by Euclid that there is an infinity of primes," he wrote. "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 said. 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, 123 = 1728, are each permutations of 2187!
However, to those inclined to seeing meaning in certain numbers, Dr. Matrix issued the following warning: "Every number has endless unusual properties."
Originally posted September 8, 1997
No comments:
Post a Comment