Sierpiński number

Template:Short description In number theory, a Sierpiński number is an odd natural number k such that <math>k \times 2^n + 1 </math> 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:

<math>\left\{\, k \cdot 2^n + 1 : n \in\mathbb{N}\,\right\}.</math>

If the form is instead <math>k \times 2^n - 1 </math>, then k is a Riesel number.

Known Sierpiński numbersEdit

The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ... (sequence A076336 in the OEIS).

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form Template:Nowrap have a factor in the covering set Template:Math}. For another known Sierpiński number, 271129, the covering set is Template:Math}. Most currently known Sierpiński numbers possess similar covering sets.<ref name="PG">Sierpinski number at The Prime Glossary</ref>

However, in 1995 A. S. Izotov showed that some fourth powers could be proved to be Sierpiński numbers without establishing a covering set for all values of n. His proof depends on the aurifeuillean factorization Template:Math. This establishes that all Template:Math give rise to a composite, and so it remains to eliminate only Template:Math using a covering set.<ref name="FQ">Template:Cite journal</ref>

Sierpiński problemEdit

Template:Unsolved

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.<ref>Template:Cite journal</ref> No smaller Sierpiński numbers have been discovered, and it is now believed that 78,557 is the smallest number.<ref>Template:Cite book</ref>

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 Template:Math is prime.<ref name="PG" /> The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

k = 21181, 22699, 24737, 55459, and 67607.

Prime Sierpiński problemEdit

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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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Extended Sierpiński problemEdit

Template:Unsolved

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 <math>78557 < k < 271129</math>. 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Simultaneously Sierpiński and RieselEdit

A number that is both Sierpiński and Riesel is a Brier number (after Éric Brier). The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, and 17855036657007596110949 (A076335).<ref>Problem 29.- Brier Numbers</ref>