From Wikipedia, the free encyclopedia - View original article

This article is about the mathematical concept. For other uses, see Lucky number (disambiguation).

Not to be confused with Fortunate number.

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 *L*_{n} denotes the *n*-th lucky number, and *p*_{n} the *n*-th prime, then *L*_{n} > *p*_{n} 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

**^**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.

- 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.