Proth number
Encyclopedia
In number theory
, a Proth number, named after the mathematician François Proth
, is a number of the form
where
is an odd positive integer
and
is a positive integer such that 
. Without the latter condition, all odd integers greater than 1 would be Proth numbers.
The first Proth numbers are :
The Cullen number
s (n·2n+1) and Fermat numbers (22n+1) are special cases of Proth numbers.
. The first Proth primes are :
The primality of a Proth number can be tested with Proth's theorem
which states that a Proth number
is prime if and only if there exists an integer 
for which the following is true:
The largest known Proth prime is
. It was found by Konstantin Agafonov in the Seventeen or Bust
distributed computing project which announced it 5 May 2007. It is also the largest known non-Mersenne prime
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a Proth number, named after the mathematician François Proth
François Proth
François Proth was a French self-taught mathematician farmer who lived in Vaux-devant-Damloup near Verdun, France. He stated four primality-related theorems, the most famous of which is Proth's theorem, published around 1878...
, is a number of the form
where
is an odd positive integerInteger
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...
and
is a positive integer such that . Without the latter condition, all odd integers greater than 1 would be Proth numbers.The first Proth numbers are :
- 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225
The Cullen number
Cullen number
In mathematics, a Cullen number is a natural number of the form n · 2n + 1 . Cullen numbers were first studied by Fr. James Cullen in 1905...
s (n·2n+1) and Fermat numbers (22n+1) are special cases of Proth numbers.
Proth primes
A Proth prime is a Proth number which is primePrime 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...
. The first Proth primes are :
- 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857.
The primality of a Proth number can be tested with Proth's theorem
Proth's theorem
In number theory, Proth's theorem is a primality test for Proth numbers.It states that if p is a Proth number, of the form k2n + 1 with k odd and k In number theory, Proth's theorem is a primality test for Proth numbers....
which states that a Proth number
is prime if and only if there exists an integer for which the following is true:The largest known Proth prime is
. It was found by Konstantin Agafonov in the Seventeen or BustSeventeen 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:...
distributed computing project which announced it 5 May 2007. It is also the largest known non-Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
.
See also
- Sierpinski number
- PrimeGridPrimeGridPrimeGrid 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...
– a distributed computing project searching for large Proth primes