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|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:
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