Sierpinski number
A Sierpinski number (named after the Polish mathematician Waclaw Sierpiński ) is a natural, odd number whose sequence of numbers of the form with no primes contains.
Counterexample
The number is not a Sierpinski number, as in the sequence, at least one prime occurs: 39, 77, 153, 305, 609, 1217, ...
Such emerging prime number is called Prothsche prime.
Sierpinski problem
The Sierpinski problem is: what is the smallest Sierpinski number? . 1967 John L. Selfridge has shown that 78 557 is a Sierpinski number. However, it is not yet known whether 78557 is the smallest Sierpinski number. But it is believed that it is the smallest number Sierpinski.
To complete the proof, a number must be found for each smaller than 78 557, so that the resulting Proth number is prime. This proof is (as of 03/2010) already occurs for all but 6 numbers: 10,223, 21,181, 22,699, 24,737, 55,459 and 67,607.
Riesel number
A trickle number (named after the Swedish mathematician Hans Riesel ) is a natural, odd number whose sequence of numbers of the form with no primes contains.
Counterexample
The number is not a Riesel number, as in the sequence occurs at least a prime number: 45, 91, 183, 367
The smallest Riesel number
Riesel found himself in 1956 with 509 203 a Riesel number. However, it is not yet known whether 509,203 is the smallest number of downflow.
Brier number
By Eric Brier k was searched for numbers that are Sierpinski and Riesel number simultaneously, ie
Are for all n always composed. Such numbers are called Brier numbers.
The first found 1998 Brier number is the 41ziffrige
Yves Gallot identified 2000 27stellige Brier number
2007 found Michael Filaseta, Carrie Finch and Mark Kozek the smallest known at the moment Brier number
The smallest known Brier numbers are listed in The On-Line Encyclopedia of Integer Sequences: List of the smallest known Brier numbers.