Sierpinski number

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In number theory, a Sierpinski number is an odd natural number k such that integers of the form k2ⁿ + 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:

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 1 Known Sierpinski numbers 2 The Sierpinski problem 3 See also 4 References 5 External links

[edit] Known Sierpinski numbers

The sequence of currently known Sierpinski numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, … (sequence A076336 in OEIS)

The number 78,557 was proved to be a Sierpinski number by John Selfridge in 1962, who showed that all numbers of the form 78557·2ⁿ+1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpinski number, 271129, the covering set is {3,5,7,13,17,241}. All currently known Sierpinski numbers possess similar covering sets.^[1]

[edit] The Sierpinski problem

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

In 1967, Sierpiński and Selfridge conjectured 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, for every odd k below 78,557 there exists an n such that k2ⁿ+1 is prime.^[2] As of June 2009, there are only six candidates:

k = 10223, 21181, 22699, 24737, 55459, and 67607

which have not been eliminated as possible Sierpinski numbers.^[3] 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.

[edit] See also

• Riesel number

[edit] References

• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 120. ISBN 0-387-20860-7. 

[edit] External links

• The Sierpinski problem: definition and status
• Sierpinski's Composite Number Theorem at MathWorld
• The Prime Sierpinski Problem, a related question.