Massacre among the integers
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The sieve of Eratosthenes works by striking out every multiple of each integer except the integer itself from a table containing the integers from 2 to N. This produces the list of prime numbers less than or equal to N.
In the 1950s, Stanislas Ulam and several colleagues proposed a rather similar little game. Start with the sequence of nonzero natural numbers. Delete every other term, leaving only the odd integers. The second number not yet deleted is 3, so delete every third term among the survivors (5, 11, 17…). The third number not yet deleted is now 7, so repeat the process, this time deleting every seventh term among the survivors (19, 39, 61…). The next "survivor" is 9, so every ninth term will be deleted, and so on.
The numbers that manage to survive this terrifying elimination process are called lucky. The first lucky numbers are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49… There are infinitely many of them, and they share properties similar to those of prime numbers. For example, their distribution among the natural numbers appears to match that of prime numbers. Goldbach's conjecture, which states that every even number is the sum of two prime numbers, is thought to remain valid for lucky numbers. Ultimately, all this raises questions about the fundamental nature of prime numbers: are their properties linked to the fact that they can be constructed using a sieve?
It is still not known whether infinitely many numbers are both prime and lucky…
