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HandWiki. Sierpinski Number. Encyclopedia. Available online: https://encyclopedia.pub/entry/37744 (accessed on 13 April 2024).

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In number theory, a Sierpiński number is an odd natural number k such that [math]\displaystyle{ 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: If the form is instead [math]\displaystyle{ k \times 2^n - 1 }[/math], then k is a Riesel number.

natural number
natural numbers
riesel

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 78557⋅2^{n} + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. Most currently known Sierpiński numbers possess similar covering sets.^{[1]}

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 *t*^{4}⋅2^{4m+2} + 1 = (*t*^{2}⋅2^{2m+1} + *t*⋅2^{m+1} + 1)⋅(*t*^{2}⋅2^{2m+1} - *t*⋅2^{m+1} + 1). This establishes that all *n* ≡ 2 (mod 4) give rise to a composite, and so it remains to eliminate only *n* ≡ 0, 1, 3 (mod 4) using a covering set.^{[2]}

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.^{[3]} No smaller Sierpiński numbers have been discovered, and it is now believed that 78,557 is the smallest number.^{[4]}

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 *k*2^{n} + 1 is prime.^{[1]} (As of November 2018), there are only five candidates which have not been eliminated as possible Sierpiński numbers:^{[5]}

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

The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of *k*. (As of February 2020), no prime has been found for these values of *k* with [math]\displaystyle{ n \le 31\,875\,742 }[/math].^{[6]}

The most recently eliminated candidate was *k* = 10223, when the prime [math]\displaystyle{ 10223\times2^{31172165} + 1 }[/math] was discovered by PrimeGrid in October 2016. This number is 9,383,761 digits long.^{[5]}

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 November 2018), the nine prime values of *k* less than 271129 for which a prime of the form *k*2^{n} + 1 is not known are:^{[7]}

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

(As of November 2019), no prime has been found for these values of *k* with [math]\displaystyle{ n \le 21\,874\,934 }[/math].^{[8]}

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 [math]\displaystyle{ 168451\times2^{19375200} + 1 }[/math] was discovered by PrimeGrid in September 2017. The number is 5,832,522 digits long.^{[9]}

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]\displaystyle{ 78557 \lt k \lt 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.

Solving the extended Sierpiński problem, the most demanding of the three posed problems, requires the elimination of 23 remaining candidates [math]\displaystyle{ k \lt 271129 }[/math], of which nine are prime (see above) and fourteen are composite. The latter include *k* = 21181, 24737, 55459 from the original Sierpiński problem, unique to the extended Sierpiński problem. (As of December 2019), the following nine values of *k* remain:^{[10]}

*k*= 91549, 131179, 163187, 200749, 202705, 209611, 227723, 229673, and 238411.

(As of September 2019), no prime has been found for these values of *k* with [math]\displaystyle{ n \le 13\,081\,160 }[/math].^{[11]}

In April 2018, [math]\displaystyle{ 193997\times2^{11452891} + 1 }[/math] was found to be prime by PrimeGrid, eliminating k = 193997. The number is 3,447,670 digits long.^{[12]}

The most recent elimination was in December 2019, when [math]\displaystyle{ 99739\times2^{14019102} + 1 }[/math] was found to be prime by PrimeGrid, eliminating k = 99739. The number is 4,220,176 digits long.^{[13]}

A number may be simultaneously Sierpiński and Riesel. These are called Brier numbers. The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... (A076335).^{[14]}

If we take *n* to be a negative integer, then the number *k*2^{n} + 1 becomes [math]\displaystyle{ \frac{2^{|n|} + k}{2^{|n|}} }[/math]. When *k* is odd, this is a fraction in reduced form, with numerator 2^{|n|} + *k*. A **dual Sierpinski number** is defined as an odd natural number *k* such that 2^{n} + *k* is composite for all natural numbers *n*. There is a conjecture that the set of these numbers is the same as the set of Sierpinski numbers; for example, 2^{n} + 78557 is composite for all natural numbers *n*.

For odd values of *k* the least *n* such that 2^{n} + *k* is prime are

- 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 5, 2, ... (sequence A067760 in the OEIS)

The odd values of *k* for which 2^{n} + *k* is composite for all *n* < *k* are

- 773, 2131, 2491, 4471, 5101, 7013, 8543, 10711, 14717, 17659, 19081, 19249, 20273, 21661, 22193, 26213, 28433, ... (sequence A033919 in the OEIS)

- Sierpinski number at The Prime Glossary http://primes.utm.edu/glossary/page.php?sort=SierpinskiNumber
- Anatoly S. Izotov (1995). "Note on Sierpinski Numbers". Fibonacci Quarterly 33 (3): 206. http://www.fq.math.ca/Scanned/33-3/izotov.pdf.
- Erdös, Paul; Odlyzko, Andrew Michael (May 1, 1979). "On the density of odd integers of the form (p − 1)2−n and related questions" (in English). Journal of Number Theory 11 (2): 257–263. doi:10.1016/0022-314X(79)90043-X. ISSN 0022-314X. https://dx.doi.org/10.1016%2F0022-314X%2879%2990043-X
- Guy, Richard Kenneth (2005) (in English). Unsolved problems in number theory. New York: Springer. pp. B21:119–121, F13:383–385. ISBN 978-0-387-20860-2. OCLC 634701581. http://www.worldcat.org/oclc/634701581
- Seventeen or Bust at PrimeGrid. http://primegrid.com/forum_thread.php?id=1647
- "Seventeen or Bust statistics". https://www.primegrid.com/stats_sob_llr.php.
- Goetz, Michael (July 10, 2008). "About the Prime Sierpinski Problem". https://www.primegrid.com/forum_thread.php?id=972.
- "Prime Sierpinski Problem statistics". https://www.primegrid.com/stats_psp_llr.php.
- Zimmerman, Van (September 29, 2017). "New PSP Mega Prime!". PrimeGrid. https://www.primegrid.com/forum_thread.php?id=7636.
- Goetz, Michael (6 April 2018). "Welcome to the Extended Sierpinski Problem". https://www.primegrid.com/forum_thread.php?id=5758.
- "Extended Sierpinski Problem statistics". https://www.primegrid.com/stats_esp_llr.php.
- Zimmerman, Van (5 April 2018). "ESP Mega Prime!". https://www.primegrid.com/forum_thread.php?id=7981.
- Brown, Scott (13 January 2020). "ESP Mega Prime!". https://www.primegrid.com/forum_thread.php?id=8982.
- Problem 29.- Brier Numbers http://www.primepuzzles.net/problems/prob_029.htm

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