Riesel number
Encyclopedia
In mathematics
, a Riesel number is an odd
natural number
k for which the integers of the form k·2n − 1 are composite
for all natural numbers n.
In other words, when k is a Riesel number, all members of the following set are composite:
In 1956, Hans Riesel
showed that there are an infinite number of integers k such that k·2n − 1 is not prime
for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer
multiple of 11184810.
A number can be shown to be a Riesel number by exhibiting a covering set
: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, 57 values of k less than this have yielded only composite numbers for all values of n so far tested. The smallest of these are 2293, 9221, 23669, 31859, 38473, 40597, 46663, 67117, 74699 and 81041.
Thirty-one numbers have had primes found by the Riesel Sieve project
(analogous to Seventeen or Bust
for Sierpinski numbers). Currently PrimeGrid
is working on remaining numbers and has found seven primes .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Riesel number is an odd
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
k for which the integers of the form k·2n − 1 are composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
for all natural numbers n.
In other words, when k is a Riesel number, all members of the following set are composite:
In 1956, Hans Riesel
Hans Riesel
Hans Ivar Riesel is a Swedish mathematician who discovered the 18th known Mersenne prime in 1957, using the computer BESK. This prime is 23217-1, which consists of 969 digits. He held the record for the largest known prime from 1957 to 1961, when Alexander Hurwitz discovered a larger one. Riesel...
showed that there are an infinite number of integers k such that k·2n − 1 is not prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
multiple of 11184810.
A number can be shown to be a Riesel number by exhibiting a covering set
Covering set
In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set...
: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
- 509203×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
- 762701×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
- 777149×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
- 790841×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
- 992077×2n − 1 has covering set {3, 5, 7, 13, 17, 241}.
The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, 57 values of k less than this have yielded only composite numbers for all values of n so far tested. The smallest of these are 2293, 9221, 23669, 31859, 38473, 40597, 46663, 67117, 74699 and 81041.
Thirty-one numbers have had primes found by the Riesel Sieve project
Riesel Sieve
Riesel Sieve is a distributed computing project, running in part on the BOINC platform. Its aim is to prove that is the smallest Riesel number, by finding a prime of the form for all odd smaller than .-Progress of the project:...
(analogous to Seventeen or Bust
Seventeen or Bust
Seventeen or Bust is a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.-Goals:...
for Sierpinski numbers). Currently PrimeGrid
PrimeGrid
PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...
is working on remaining numbers and has found seven primes .