If you have to play games of chance involving numbers, why not stick with the lucky numbers? That is why not choose the numbers among the set {1,3,7,9,13,15,21, 25,31,33,37,43,49,51,63,67,69...}. If you are looking for a pattern, you might be hard pressed to find one, although you will shortly see how these lucky numbers are generated. What is interesting about this set of numbers is that they share some very important properties displayed by primes, and what remains unknown about them is whether the subset of lucky primes---that is the lucky numbers which are primes---is infinite.
Sieve of Eratosthenes
The sieve of Eratosthenes was created by a third century Greek mathematician and named eponymously. This sieve effectively filters out the prime numbers from a list of consecutive integers up to a predetermined number. The way the sieve works is as follows (illustrated here for primes less than 50): write down the numbers 2 to 50 and then successively cross out all multiples of 2, then 3, then 5, and so on. What is left is the set {2,3,5,7,11,13,17,19,23,29,31,37,39,41,43,47}.
This sieve is an effective, albeit somewhat tedious, way of finding primes up to some preset number. Once the list of numbers are "passed through" utilizing the successive divisors, what remains is guaranteed prime. The list is a useful way for children to create a list of primes up to a specific number, all the while getting great practice with division and number multiples. If a square matrix or tabular array of the numbers is created, then the manner in which the primes fall out is observed and the occurrence of such curiosities as prime twins and triplets is noted.
Lucky Numbers
In like manner, the lucky numbers can be generated by a number sieve, with the rule for selecting the numbers changed somewhat. To begin, list the numbers from 1 to some predetermined limit, let us say, 100. Unlike the sieve of Eratosthenes, in which we strike out multiples, to generate the lucky numbers, we strike out positional numbers. That is, we start by striking out every other number in the list. This leaves the odds, 1,3,5,7,9... We then take the second number on the list 3, and begin eliminating every third number on the list. This leaves 1,3,7,9,13,15,17,19... The third surviving number on the list is 7. We thus start eliminating every seventh number on the list. We continue in this fashion until we have the survivors, or lucky numbers: 1,3,7,9,13,15,21,25,31,33,37,43,49,51,63...
What is curious from the list above is that 7 is indeed a lucky number. Maybe this is why throughout history the number 7 has been considered lucky and a favorite of many. Yet 13 is also on the list and many regard this as an unlucky number, to wit "Friday the thirteenth." Perhaps to determine whether these numbers truly are lucky or not will require them to be implemented in games of chance and lottery type games. The results will then tell us how they fared. Whether your experience with these numbers turns out to be lucky, or unlucky, they remain a curious list indeed.
References
- Clawson, Calvin C. Mathematical Mysteries: The Beauty and Magic of Numbers. New York: Basic Books, 1999.
Joe Pagano - Joe is a prolific writer, author, poet, and linguist, who has published over 200 articles and several books on a broad array of ...
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