A Sierpiński number of the first kind is a number of the form . The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if is prime with , then must be of the form , making a Fermat number with . The first few of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127).
The numbers of digits in the number is given by
where 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 . The status of Sierpiński numbers is summarized in the table below (Nielsen).
See alsoCullen Number,
Cunningham Number,
Fermat Number,
Sierpiński Number of the Second Kind,
Woodall Number Explore with Wolfram|Alpha ReferencesKeller, W. "Factors of Fermat Numbers and Large Primes of the Form ." Math. Comput. 41, 661-673, 1983.Keller, W. "Factors of Fermat Numbers and Large Primes of the Form , II." In prep.Keller, W. "Prime Factors of Fermat Numbers 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. "." 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|AlphaSierpiński 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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