Lucky number

Encyclopedia : L : LU : LUC : Lucky number

This article is about the formal mathematical concept. See Numerology for a discussion of the more common meaning.

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.

We 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,

Then we cross out every second number (all even numbers), 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. Now we cross out every third number which remains in the list:

1,    3,          7,    9,         13,   15,         19,   21,         25,

The third surviving number is now 7 so we cross out every seventh number that remains:

1,    3,          7,    9,         13,   15,               21,         25,

If we repeat this procedure 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, ...

Stanisław Ulam was the first to discuss these numbers, around 1955. He named them "lucky" because of a connection with a story told by the historian Josephus.

Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. It is not known whether there are also infinitely many lucky primes:

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, ...

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.

External links

Peterson, Ivars: MathTrek: Martin Gardner's Lucky Number: http://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm
Schneider, Walter: A list of the first 1000 lucky numbers: http://www.wschnei.de/number-theory/lucky-numbers-list.html
Sloane, Neil J. A.: A sequence of lucky numbers - A000959: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000959
Sloane, Neil J. A.: A sequence of lucky primes - A031157: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=031157
Sloane, Neil J. A.: A sequence of composite lucky numbers - A031879: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=031879
Weisstein, Eric W.: Lucky Number: http://mathworld.wolfram.com/LuckyNumber.html

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