Proth number
In number theory, a Proth number is a number of the form
where is an odd positive integer and is a positive integer such that . They are named after the mathematician François Proth. The first few Proth numbers are
- 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241 (sequence A080075 in the OEIS).
The Cullen numbers (numbers of the form n·2n + 1) and Fermat numbers (numbers of the form 22n + 1) are special cases of Proth numbers. Without the condition that , all odd integers greater than 1 would be Proth numbers.[1]
Proth primes
![]() | Unsolved problem in mathematics: Are there infinitely many Proth primes? (more unsolved problems in mathematics) |
A Proth prime is a Proth number which is prime. The first few 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 (
A080076).
The primality of a Proth number can be tested with Proth's theorem, which states[2] that a Proth number is prime if and only if there exists an integer for which
The largest known Proth prime as of 2016 is , and is 9,383,761 digits long.[3] It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016.[4] It is also the largest known non-Mersenne prime.[5]
See also
- Sierpinski number
- Pierpont Primes
- PrimeGrid – a distributed computing project searching for large Proth primes
References
- ↑ Weisstein, Eric W. "Proth Number". MathWorld.
- ↑ Weisstein, Eric W. "Proth's Theorem". MathWorld.
- ↑ Caldwell, Chris. "The Top Twenty: Proth". The Prime Pages.
- ↑ Van Zimmerman (30 Nov 2016) [9 Nov 2016]. "World Record Colbert Number discovered!". PrimeGrid.
- ↑ Caldwell, Chris. "The Top Twenty: Largest Known Primes". The Prime Pages.
External links
Grime, Dr. James. "78557 and Proth Primes" (video). YouTube. Brady Haran. Retrieved 13 November 2017.