Editing Lucky number (section)

{{Short description|Integer filtered out using a sieve similar to that of Eratosthenes}}
{{About|the mathematical concept}}
{{Distinguish|Fortunate number}}

In [[number theory]], a '''lucky number''' is a [[natural number]] in a set which is generated by a certain "[[sieve theory|sieve]]". This sieve is similar to the [[sieve of Eratosthenes]] that generates the [[prime number|primes]], but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).<ref>{{Cite web|last=Weisstein, Eric W.|title=Lucky Number|url=https://mathworld.wolfram.com/LuckyNumber.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref>

The term was introduced in 1956 in a paper by Gardiner, Lazarus, [[Nicholas Metropolis|Metropolis]] and [[Stanislaw Ulam|Ulam]].  In the same work they also suggested calling another sieve, "the sieve of [[Josephus]] Flavius"<ref>{{cite journal | zbl=0071.27002 | last1=Gardiner | first1=Verna | last2=Lazarus | first2=R. | last3=Metropolis | first3=N. | author3-link=Nicholas Metropolis | last4=Ulam | first4=S. | author4-link=Stanislaw Ulam | title=On certain sequences of integers defined by sieves | journal=[[Mathematics Magazine]] | volume=29 | pages=117–122 | year=1956 | issn=0025-570X | doi=10.2307/3029719 | issue=3 | jstor=3029719 }}</ref> 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.  Twin lucky numbers and [[twin prime]]s also appear to occur with similar frequency.  However, if ''L''<sub>''n''</sub> denotes the ''n''-th lucky number, and ''p''<sub>''n''</sub> the ''n''-th prime, then ''L''<sub>''n''</sub> > ''p''<sub>''n''</sub> for all sufficiently large ''n''.<ref>{{cite journal | zbl=0084.04202 | last1=Hawkins | first1=D. | last2=Briggs | first2=W.E. | title=The lucky number theorem | journal=[[Mathematics Magazine]] | volume=31 | pages=81–84,277–280 | year=1957 | issn=0025-570X | doi=10.2307/3029213 | issue=2 | jstor=3029213 }}</ref>

Because of their apparent similarities with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this [[conjecture]].