proth number

Prothsche primes are primes of the form, consistent with and odd ( such numbers generally glad Proth numbers, even if they are not primes ). Each prime number can be uniquely written in the form; thus a prime number is a prime number Prothsche, but must also apply.

The importance of Prothschen primes is that Proth François (1852-1879) has a simple test found ( set of Proth ), which can be detected whether Proth numbers are prime. Many of the currently largest known primes have been found with this test, and there is a freely available program by Yves Gallot is that it implements the set of Proth and often used for such purposes.

The set of states Proth: The Proth number N is prime if there exists a natural number a is with:

The Prothschen primes play in the Sierpinski numbers so far as a role, as a sequence of numbers of the form must be free of Prothschen primes, so that a Sierpinski number may be.

Among the prothschen primes are also Cullen primes. These are primes of the form.

In the following table, find prime numbers ordered by. Primes, which are therefore not Prothschen primes are in parentheses.

The largest known prime number Proth ( Konstantin Agafonov 2007).