TOPICS
Search

Sierpiński Number of the First Kind

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

A Sierpiński number of the first kind is a number of the form S_n=n^n+1. The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if S_n is prime with n>=2, then n must be of the form n=2^(2^k), making S_n a Fermat number F_m with m=k+2^k. The first few m of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127).

The numbers of digits in the number S_k is given by

d_k=[2^(k+2^k)log_(10)2],

where [z] is the ceiling function, so the numbers of digits in the first few candidates are 1, 3, 20, 617, 315653, 41373247568, ... (OEIS A089943).

The only known prime Sierpiński numbers of the first kind are 2, 5, 257, with the first unknown case being F_(70)>10^(3×10^(20)). The status of Sierpiński numbers is summarized in the table below (Nielsen).

k m status of F_m=S(n)
0 1 prime (S_n=5)
1 3 prime (S_n=257)
2 6 composite with factor 1071·2^8+1
3 11 composite with factor 39·2^(13)+1
4 20 composite with no factor known
5 37 composite with factor 1275438465·2^(39)+1
6 70 unknown
7 135 unknown
8 264 unknown
9 521 unknown
10 1034 unknown
11 2059 composite with factor 591909·2^(2063)+1
12 4108 unknown
13 8205 unknown
14 16398 unknown
15 32783 unknown
16 65552 unknown
17 131089 unknown

See also

Cullen Number, Cunningham Number, Fermat Number, Sierpiński Number of the Second Kind, Woodall Number

Explore with Wolfram|Alpha

References

Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^n+1." Math. Comput. 41, 661-673, 1983.Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^n+1, II." In prep.Keller, W. "Prime Factors k·2^n+1 of Fermat Numbers F_m and Complete Factoring Status." http://www.prothsearch.net/fermat.html.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 155, 1979.Nielsen, J. S. "n^n+1." http://jeppesn.dk/nton.html.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 74, 1989.Sloane, N. J. A. Sequences A006127/M2547, A014566, A089943 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Sierpiński Number of the First Kind

Cite this as:

Weisstein, Eric W. "Sierpiński Number of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SierpinskiNumberoftheFirstKind.html

Subject classifications