# Sierpinski number

In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that ${\displaystyle k\times 2^{n}+1}$ 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:

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

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

## Known Sierpiński numbers

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⋅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}. 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 t4⋅24n+2 + 1 = (t2⋅22n+1 + t⋅2n+1 + 1)⋅(t2⋅22n+1 - t⋅2n+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]

## Sierpiński problem

 Unsolved problem in mathematics:Is 78,557 the smallest Sierpiński number?(more unsolved problems in mathematics)

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 August 2017, there are only five candidates:

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

which have not been eliminated as possible Sierpiński numbers.[3]

## Extended Sierpiński problem

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 ${\displaystyle 78557. A 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 ${\displaystyle k<271129}$, 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 April 2018, the following ten values of k remain:

k = 91549, 99739, 131179, 163187, 200749, 202705, 209611, 227723, 229673, 238411.

The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k. As of April 2018, no prime has been found for these values of k with ${\displaystyle n<11\,313\,676}$.[4]

The most recent elimination was in April 2018, when ${\displaystyle 193997\times 2^{11452891}+1}$ was found to be prime by PrimeGrid, eliminating k=193997. The number is 3,447,670 digits long.[5]

## Simultaneously Sierpiński and Riesel

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).[6]

## Dual Sierpinski problem

If we take the n of k2n + 1 to a negative integer, then the number become ${\displaystyle {\frac {2^{-n}+k}{2^{-n}}}}$. If we choose the numerator, then the number become 2n + k. Thus, a dual Sierpinski number is defined as an odd natural number k such that 2n + 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, 2n + 78557 is composite for all natural numbers n.

The least n such that 2n + k is prime are (for odd ks)

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, 1, 3, 2, 1, 1, 8, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 7, 2, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 7, 4, 5, 3, 4, 2, 1, 2, 1, 3, 2, 1, 1, 10, 3, 3, 2, 1, 1, ... (sequence A067760 in the OEIS)

The odd ks which 2n + k are 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)

There is also a "five or bust", similar to seventeen or bust, considers this problem, and found (probable) primes for all k < 78557 (the largest prime is 29092392 + 40291[7]), so it is currently known that 78557 is the smallest dual Sierpinski number.