In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.
| Begin with a list of integers starting with 1: | ||||||||||||||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
| Every second number (all even numbers) is eliminated, leaving only the odd integers: | ||||||||||||||||||||||||
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | ||||||||||||
| The second term in this sequence is 3. Every third number which remains in the list is eliminated: | ||||||||||||||||||||||||
| 1 | 3 | 7 | 9 | 13 | 15 | 19 | 21 | 25 | ||||||||||||||||
| The next surviving number is now 7, so every seventh number that remains is eliminated: | ||||||||||||||||||||||||
| 1 | 3 | 7 | 9 | 13 | 15 | 21 | 25 | |||||||||||||||||
When this procedure has been carried out completely, the survivors are the lucky numbers:
- 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... (sequence A000959 in OEIS).
The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius"[1] because of its similarity with the counting-out game in the Josephus problem.
Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. However, if Ln denotes the n-th lucky number, and pn the n-th prime, then Ln > pn for all sufficiently large n.[2]
Because of these apparent connections with the prime numbers, some mathematicians have suggested that these properties may be found in a larger class of sets of numbers generated by sieves of a certain unknown form, although there is little theoretical basis for this conjecture. Twin lucky numbers and twin primes also appear to occur with similar frequency.
A lucky prime is a lucky number that is prime. It is not known whether there are infinitely many lucky primes. The first few are
References[edit]
- ^ Gardiner et al (1956)
- ^ Hawkins, D.; Briggs, W.E. (1957). "The lucky number theorem". Mathematics Magazine 31 (2): 81–84,277–280. doi:10.2307/3029213. ISSN 0025-570X. Zbl 0084.04202.
- Gardiner, Verna; Lazarus, R.; Metropolis, N.; Ulam, S. (1956). "On certain sequences of integers defined by sieves". Mathematics Magazine 29 (3): 117–122. doi:10.2307/3029719. ISSN 0025-570X. Zbl 0071.27002.
- Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. C3. ISBN 978-0-387-20860-2. Zbl 1058.11001.
External links[edit]
- Peterson, Ivars. MathTrek: Martin Gardner's Lucky Number
- Weisstein, Eric W., "Lucky Number", MathWorld.
- Lucky Numbers by Enrique Zeleny, The Wolfram Demonstrations Project.
- Symonds, Ria. "31: And other lucky numbers". Numberphile. Brady Haran.