In number theory, a Sierpià Âski number is an odd natural number k such that is composite for all natural numbers n. In 1960, Wacà Âaw Sierpià Âski proved that there are infinitely many odd integers k which have this property.
In other words, when k is a Sierpià Âski number, all members of the following set are composite:
If the form is instead , then k is a Riesel number.
The sequence of currently known Sierpià Âski numbers begins with:
The number 78557 was proved to be a Sierpià Âski number by John Selfridge in 1962, who showed that all numbers of the form have a factor in the covering set }. For another known Sierpià Âski number, 271129, the covering set is }. Most currently known Sierpià Âski numbers possess similar covering sets.
However, in 1995 A. S. Izotov showed that some fourth powers could be proved to be Sierpià Âski numbers without establishing a covering set for all values of n. His proof depends on the aurifeuillean factorization . This establishes that all give rise to a composite, and so it remains to eliminate only using a covering set.
The Sierpià Âski problem asks for the value of the smallest Sierpià Âski number. In private correspondence with Paul Erdà Âs, Selfridge conjectured that 78,557 was the smallest Sierpià Âski number. No smaller Sierpià Âski numbers have been discovered, and it is now believed that 78,557 is the smallest number.
To show that 78,557 really is the smallest Sierpià Âski number, one must show that all the odd numbers smaller than 78,557 are not Sierpià Âski numbers. That is, for every odd k below 78,557, there needs to exist a positive integer n such that is prime. The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:
The current status for the remaining multipliers can be seen at PrimeGrid's website.
In 1976, Nathan Mendelsohn determined that the second provable Sierpià Âski number is the prime k = 271129. The prime Sierpià Âski problem asks for the value of the smallest prime Sierpià Âski number, and there is an ongoing "Prime Sierpià Âski search" which tries to prove that 271129 is the first Sierpià Âski number which is also a prime.
Suppose that both preceding Sierpià Âski problems had finally been solved, showing that 78557 is the smallest Sierpià Âski number and that 271129 is the smallest prime Sierpià Âski number. This still leaves unsolved the question of the second Sierpinski number; there could exist a composite Sierpià Âski number k such that . An ongoing search is trying to prove that 271129 is the second Sierpià Âski number, by testing all k values between 78557 and 271129, prime or not.
A number that is both Sierpià Âski and Riesel is a Brier number (after ÃÂric Brier). The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, and 17855036657007596110949 (); it is not known whether other Brier numbers smaller than these exist (i.e., they may not be the five smallest).