Sierpiński number

In number theory, a Sierpiński number is an odd natural number k such that

k

×

2

n

+

1

{\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:

{

k

⋅

2

n

+

1

:

n

∈

N

}

.

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

If the form is instead

k

×

2

n

−

1

{\displaystyle k\times 2^{n}-1}

, then k is a Riesel number.

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.

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.