Sierpinski Number of the First Kind

Numbers of the form $S_n\equiv n^n+1$. The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (Sloane's A014566). Sierpinski proved that if $S_n$ is Prime with $n\geq 2$, then $S_n=F_{m+2^m}$, where $F_m$ is a Fermat Number with $m\geq 0$. The first few such numbers are $F_1=5$, $F_3=257$, $F_6$, $F_{11}$, $F_{20}$, and $F_{37}$. Of these, 5 and 257 are Prime, and the first unknown case is $F_{37}>10^{3\times 10^{10}}$.

See also Cullen Number, Cunningham Number, Fermat Number, Woodall Number

References

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 155, 1979.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.

Sloane, N. J. A. Sequence A014566 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.