Sierpinski number

In number theory, a Sierpinski number is an odd natural number "k" such that integers of the form "k"2^"n" + 1 are composite (i.e. not prime) for all natural numbers "n".

In other words, when "k" is a Sierpinski number, all members of the following set are composite:

:$left\{,k\; 2^n\; +\; 1\; :\; n\; inmathbb\{N\},\; ight\}.$

Numbers in this set with odd k and k < 2ⁿ are called Proth numbers.

In 1960 Wacław Sierpiński proved that there are infinitely many odd integers that when used as "k" produce no primes.

The Sierpinski problem

The Sierpinski problem is: "What is the smallest Sierpinski number?"

In 1962, John Selfridge proved that 78,557 is a Sierpinski number; he showed that, when "k"=78,557, all numbers of the form "k"2^"n"+1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}.

In addition, in 1967, Sierpiński and Selfridge proposed (but could not prove) the conjecture that 78,557 is the smallest Sierpinski number, and thus the answer to the Sierpinski problem.

To show that 78,557 really is the smallest Sierpinski number, one must show that all the odd numbers smaller than 78,557 are "not" Sierpinski numbers. That is, there exists an "n" such that "k"2^"n"+1 is prime. [http://primes.utm.edu/links/theory/special_forms/Sierpinski/] As of November 2007, there are only six candidates which have not been eliminated as possible Sierpinski numbers. [http://seventeenorbust.com/stats/rangeStatsEx.mhtml] Seventeen or Bust, a distributed computing project, is testing these remaining numbers.

If the project finds a prime of the right form for all the remaining "k", the Sierpinski problem will be solved.

Current status

As of September 2008, the following "k" have been solved by Seventeen or Bust.

ee also

* Seventeen or Bust
* Riesel number

References

*

External links

* [http://primes.utm.edu/glossary/page.php?sort=SierpinskiNumber Sierpinski number at The Prime Glossary]
* [http://www.prothsearch.net/sierp.html The Sierpinski problem: definition and status]
* [http://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html Sierpinski's Composite Number Theorem at MathWorld]
* [http://www.mersenneforum.org/showthread.php?t=2665 The Prime Sierpinski Problem] , a related question.