Сиерпински сандары және олардың іздеуі

In number theory, a Sierpiński number is an odd natural number k such that is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all members of the following set are composite:

If the form is instead , then k is a Riesel number.

The Sierpiński problem asks for the value of the smallest Sierpiński number. In private correspondence with Paul Erdős, Selfridge conjectured that 78,557 was the smallest Sierpiński number. No smaller Sierpiński 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 Sierpiński number, one must show that all the odd numbers smaller than 78,557 are not Sierpiński numbers. That is, for every odd k below 78,557, there needs to exist a positive integer n such that k2^(n) + 1 is prime. The most recently eliminated candidate was k = 10223, when the prime was discovered by PrimeGrid in October 2016. This number is 9,383,761 digits long.

In 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime k = 271129. The prime Sierpiński problem asks for the value of the smallest prime Sierpiński number, and there is an ongoing "Prime Sierpiński search" which tries to prove that 271129 is the first Sierpiński number which is also a prime. as of 2018, the nine prime values of k less than 271129 for which a prime of the form k2^(n) + 1 is not known are:

k = 22699, 67607, 79309, 79817, 152267, 156511, 222113, 225931, and 237019.

as of 2024, no prime has been found for these values of k with
The first two, being less than 78557, are also unsolved cases of the (non prime) Sierpiński problem described above. The most recently eliminated candidate was k = 168451, when the prime number was discovered by PrimeGrid in September 2017. The number is 5,832,522 digits long.

Suppose that both preceding Sierpiński problems had finally been solved, showing that 78557 is the smallest Sierpiński number and that 271129 is the smallest prime Sierpiński number. This still leaves unsolved the question of the second Sierpinski number; there could exist a composite Sierpiński number k such that An ongoing search is trying to prove that 271129 is the second Sierpiński number, by testing all k values between 78557 and 271129, prime or not. Solving the extended Sierpiński problem, the most demanding of the three posed problems, requires the elimination of 21 remaining candidates , of which nine are prime (see above) and twelve are composite. The latter include k = 21181, 24737, 55459 from the original Sierpiński problem. as of 2022, the following eight values of k, unique to the extended Sierpiński problem, remain:

k = 91549, 131179, 163187, 200749, 209611, 227723, 229673, and 238411.

as of 2024, no prime has been found for these values of k with
In December 2019, was found to be prime by PrimeGrid, eliminating k = 99739. The number is 4,220,176 digits long. The most recent elimination was in December 2021, when was found to be prime by PrimeGrid, eliminating k = 202705. The number is 6,418,121 digits long.

A number may be simultaneously Sierpiński and Riesel. These are called Brier numbers. The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, .