|Begin with a list of integers starting with 1:|
|Every second number (all even numbers) is eliminated, leaving only the odd integers:|
|The second term in this sequence is 3. Every third number which remains in the list is eliminated:|
|The next surviving number is now 7, so every seventh number that remains is eliminated:|
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" 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.
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
- 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.
- 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.