Sierpinski number

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In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that k2n + 1 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:

\left\{\,k 2^n + 1 : n \in\mathbb{N}\,\right\}.

Numbers in such a set with odd k and k < 2n are Proth numbers.

Known Sierpiński numbers[edit]

The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, … (sequence A076336 in OEIS).

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form 78557·2n+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}. All currently known Sierpiński numbers possess similar covering sets.[1]

The Sierpiński problem[edit]

List of unsolved problems in mathematics
Is 78,557 the smallest Sierpiński number?

The Sierpiński problem is: "What is the smallest Sierpiński number?"

In 1967, Sierpiński and Selfridge conjectured that 78,557 is the smallest Sierpiński number, and thus the answer to the Sierpiński problem.

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 exists a positive integer n such that k2n+1 is prime.[1] As of December 2013, there are only six candidates:

k = 10223, 21181, 22699, 24737, 55459, and 67607

which have not been eliminated as possible Sierpiński numbers.[2] Seventeen or Bust (with PrimeGrid), a distributed computing project, is testing these remaining numbers. If the project finds a prime of the form k2n+1 for every remaining k, the Sierpiński problem will be solved.

The smallest n for which k×2n+1 is prime: (sequence A046067 in OEIS)

k 1 3 5 7 9 11 13 15 17 19 ... 47 ... 383 ... 5359 ... 10223 ... 21181 ... 22699 ... 24737 ... 55459 ... 67607 ... 78557 ...
Smallest n 0 1 1 2 1 1 2 1 3 6 583 6393 5054502 ? ? ? ? ? ? none

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Further reading[edit]

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