Sierpiński number

In number theory, a 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:

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

If the form is instead ${\displaystyle k\times 2^{n}-1}$, then k is a Riesel number.

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⋅2^4m+2 + 1 = (t²⋅2^2m+1 + t⋅2^m+1 + 1)⋅(t²⋅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 k2ⁿ + 1 is prime.^[1] The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:^[5]

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

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.^[6]

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 ${\displaystyle 78557<k<271129}$. 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.^[7]

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).^[8]

• Cullen number
• Proth number
• Riesel number
• Seventeen or Bust
• Woodall number

• Guy, Richard K. (2004), Unsolved Problems in Number Theory, New York: Springer-Verlag, p. 120, ISBN 0-387-20860-7

• The Sierpinski problem: definition and status
• Weisstein, Eric W. "Sierpinski's composite number theorem". MathWorld.
• Archived at Ghostarchive and the Wayback Machine: Grime, Dr. James. "78557 and Proth Primes" (video). YouTube. Brady Haran. Retrieved 13 November 2017.