Sierpiński Number of the First Kind (original) (raw)

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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).

See also

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

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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."

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Sierpiński Number of the First Kind

Cite this as:

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

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