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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]

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Proth primesEdit

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 (OEISA080076).

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 alsoEdit

ReferencesEdit

  1. ^ Weisstein, Eric W. "Proth Number". MathWorld.
  2. ^ Weisstein, Eric W. "Proth's Theorem". MathWorld.
  3. ^ Caldwell, Chris. "The Top Twenty: Proth". The Prime Pages.
  4. ^ Van Zimmerman (30 Nov 2016) [9 Nov 2016]. "World Record Colbert Number discovered!". PrimeGrid.
  5. ^ Caldwell, Chris. "The Top Twenty: Largest Known Primes". The Prime Pages.

External linksEdit