- Lucky number
In
number theory , a lucky number is anatural number in a set which is generated by a "sieve" similar to theSieve of Eratosthenes that generates the primes.Begin with a list of
integer s 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 number s) 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 third surviving number is now 7, so every seventh number that remains is eliminated:
1, 3, 7, 9, 13, 15, 21, 25,As this procedure is repeated indefinitely, 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, ... OEIS|id=A000959.
The term was introduced in 1955 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve the sieve of
Josephus Flavius. [V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam, "On certain sequences of integers defined by sieves", "Mathematics Magazine" 29:3 (1955), pp. 117–122.]Lucky numbers share some properties with primes, such as asymptotic behaviour according to the
prime number theorem ; alsoGoldbach's conjecture has been extended to them. There are infinitely many lucky numbers. 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 thisconjecture . Twin lucky numbers andtwin prime s 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:3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193 OEIS|id=A031157.
References
See also
*
Fortunate number External links
* Peterson, Ivars. [http://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm MathTrek: Martin Gardner's Lucky Number]
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* [http://demonstrations.wolfram.com/LuckyNumbers/ Lucky Numbers] by Enrique Zeleny,The Wolfram Demonstrations Project .
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