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" 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
- 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193 (sequence A031157 in OEIS).
Famous quotes containing the words lucky and/or number:
“You could not tell, and yet it looked as if
The shore was lucky in being backed by cliff,
The cliff in being backed by continent;
It looked as if a night of dark intent
Was coming, and not only a night, an age.”
—Robert Frost (1874–1963)
“A good marriage ... is a sweet association in life: full of constancy, trust, and an infinite number of useful and solid services and mutual obligations.”
—Michel de Montaigne (1533–1592)