Proth number について

. 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, 241, etc.
The Cullen numbers (''n''·2^''n''+1) and Fermat numbers (2^{2^''n''}+1) are special cases of Proth numbers.
== Proth primes ==

A Proth prime is a Proth number which is prime. 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 which states that a Proth number

p

is prime if and only if there exists an integer

a

for which the following is true:
:

a^}\equiv -1\ \pmod

The largest known Proth prime is

19249 \cdot 2^ + 1

.〔Chris Caldwell, (The Top Twenty: Proth ), from The Prime Pages.〕 It was found by Konstantin Agafonov in the Seventeen or Bust distributed computing project which announced it 5 May 2007.〔(Press Release by Seventeen or Bust ). 5 May 2007.〕 It is also the largest known non-Mersenne prime.〔Chris Caldwell, (The Top Twenty: Largest Known Primes ), from The Prime Pages.〕

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