SIERPINSKI NUMBER

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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.

Contents

The Sierpinski problem

See also

External links

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.[1] As of July 2007, there are only seven candidates which have not been eliminated as possible Sierpinski numbers.[2] Seventeen or Bust, a distributed computing project, is testing these remaining numbers. On March 26, 2007, Seventeen or Bust proved that 19,249 × 2^13,018,586 + 1, a 3,918,990 digit number, is prime, thus eliminating ''k'' = 19,249 as a possible Sierpinski number.[3] If the project finds a prime of the right form for all the remaining ''k'', the Sierpinski problem will be solved.

See also

★ Riesel number

★ Proth number

External links

★ Sierpinski number at The Prime Glossary

★ The Sierpinski problem: defintion and status

★ Sierpinski's Composite Number Theorem at MathWorld

★ The Prime Sierpinski Problem, a related question.