Sierpiński Number of the First Kind
المؤلف:
Ribenboim, P.
المصدر:
The New Book of Prime Number Records. New York: Springer-Verlag
الجزء والصفحة:
...
15-12-2020
1340
Sierpiński Number of the First Kind
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 ![[z]](https://mathworld.wolfram.com/images/equations/SierpinskiNumberoftheFirstKind/Inline11.gif)
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).
 |
 |
status of  |
| 0 |
1 |
prime ( ) |
| 1 |
3 |
prime ( ) |
| 2 |
6 |
composite with factor  |
| 3 |
11 |
composite with factor  |
| 4 |
20 |
composite with no factor known |
| 5 |
37 |
composite with factor  |
| 6 |
70 |
unknown |
| 7 |
135 |
unknown |
| 8 |
264 |
unknown |
| 9 |
521 |
unknown |
| 10 |
1034 |
unknown |
| 11 |
2059 |
composite with factor  |
| 12 |
4108 |
unknown |
| 13 |
8205 |
unknown |
| 14 |
16398 |
unknown |
| 15 |
32783 |
unknown |
| 16 |
65552 |
unknown |
| 17 |
131089 |
unknown |
REFERENCES:
Keller, 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." https://www.prothsearch.net/fermat.html.
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 155, 1979.
Nielsen, J. S. "
." https://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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