Sierpiński number

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:

$\left\{\,k2^{n}+1:n\in \mathbb {N} \,\right\}.$

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, … (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}. All currently known Sierpiński numbers possess similar covering sets.

The Sierpiński problem

Further information: Seventeen or Bust

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. 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. 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 A040076 in the OEIS)

k	Smallest n
1	0
2	0
3	1
4	0
5	1
6	0
7	2
8	1
9	1
10	0
11	1
12	0
13	2
14	1
15	1
16	0
17	3
18	0
19	6
20	1
21	1
22	0
23	1
24	2
25	2
...	...
47	583
...	...
143	53
...	...
383	6393
...	...
2897	9715
...	...
3061	33288
...	...
4847	3321063
...	...
5359	5054502
...	...
10223	Unknown
...	...
19249	13018586
...	...
21181	Unknown
...	...
22699	Unknown
...	...
24737	Unknown
...	...
27653	9167433
...	...
28433	7830457
...	...
33661	7031232
...	...
44131	995972
...	...
46157	698207
...	...
54767	1337287
...	...
55459	Unknown
...	...
65567	1013803
...	...
67607	Unknown
...	...
69109	1157446
...	...
78557	None
...	...
79309	Unknown
...	...
79817	Unknown
...	...
90527	9162167
...	...
91549	Unknown
...	...
94373	3206717
...	...
98749	1045226
...	...
99739	Unknown
...	...
107929	1007898
...	...
123287	2538167
...	...
131072	Unknown
...	...
131179	Unknown
...	...
149183	1666957
...	...
152267	Unknown
...	...
156511	Unknown
...	...
161041	Unknown
...	...
161957	727995
...	...
163187	Unknown
...	...
168451	Unknown
...	...
193997	Unknown
...	...
214519	1929114
...	...
216751	903792
...	...
222113	Unknown
...	...
222361	2854840
...	...
271129	None
...	...

See also

• Riesel number

Further reading

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

External links

• The Sierpinski problem: definition and status
• Weisstein, Eric W. "Sierpinski's composite number theorem". MathWorld.
• The Prime Sierpinski Problem, a related question.