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A number satisfying Sierpinski's Composite Number Theorem, i.e.,
such that
is Composite for every
. The smallest known is
, but there remain 35
smaller candidates (the smallest of which is 4847) which are known to generate only composite numbers for
or more (Ribenboim 1996, p. 358).
Let be smallest
for which
is Prime, then the first few values are 0, 1, 1, 2, 1, 1, 2, 1,
3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, ... (Sloane's A046067). The second smallest
are given by 1, 2, 3, 4, 2,
3, 8, 2, 15, 10, 4, 9, 4, 4, 3, 60, 6, 3, 4, 2, 11, 6, 9, 1483, ... (Sloane's A046068). Quite large
can be required to
obtain the first prime even for small
. For example, the smallest prime of the form
is
. There are an infinite number of Sierpinski numbers which are Prime.
The smallest odd such that
is Composite for all
are 773, 2131, 2491, 4471, 5101, ....
See also Mersenne Number, Riesel Number, Sierpinski's Composite Number Theorem
References
Buell, D. A. and Young, J. ``Some Large Primes and the Sierpinski Problem.'' SRC Tech. Rep. 88004, Supercomputing Research
Center, Lanham, MD, 1988.
Jaeschke, G. ``On the Smallest
Jaeschke, G. Corrigendum to ``On the Smallest
Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form
Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.
Sierpinski, W. ``Sur un problème concernant les nombres
Sloane, N. J. A.
A046067 and
A046068
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.
such that
are Composite.'' Math. Comput. 40, 381-384, 1983.
such that
are Composite.'' Math. Comput. 45, 637, 1985.
.'' Math. Comput. 41, 661-673, 1983.
, II.'' In prep.
.'' Elem. d. Math. 15, 73-74, 1960.
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© 1996-9 Eric W. Weisstein