Originally, Yves Gallot wrote a program to implement the following theorem
from our pages on primality proving
Proth's Theorem (1878): Let N = k.2n+1
with 2n > k. If there is an integer
a such that
This test is so simple that in practice the difficulty is quickly
multiplying the large numbers involved. It is also a very useful test:
it applies to Cullen
factors, the primes in the Sierpinski
conjecture... (See Ray Ballinger's Proth
Range Page for additional information.)
a(N-1)/2 = -1 (mod
then N is prime.
has now expanded this program to also cover prime of the form k.2n-1!
His "Proth.exe" program makes finding all these types of primes
as easy as selecting the form you desire from a menu, choosing the starting
values (here Ray Ballinger's Proth
Range page's are especially helpful) and then letting the machine
go. Big primes take awhile to find, so Yves programs tracks where it is
and can automatically continue each time you start your machine. You can
adjust the priority of the program so it only runs when your machines
is doing nothing else...
Other features include the ability to set a range for the starting k
(odd) and n, to set certain algebraic forms for the multiplier,
and to test (automatically) if any prime you find is part of a twin
prime or Sophie
Germain prime pair...
To download, choose one of the following:
Ranges to search & Projects to join
- Paul Jobling's NewPGen which can be used
to pre-sieve candidates for Prothe.exe searches.
- George Woltman's PRP
can be used to take the output of NewPGen, apply a prp test, then create
a file for Proth.exe to prove. Get prp.zip for Windows and
prp.tgz for Linux on an Intel *86 processor.
- Chris Nash's PrimeForm
- built with the same arithmetic library as Proth.exe, PrimeForm allows
much move variety in the form of the prime numbers sought.
- Andy Penrose's Proth
benchmarks for various machines
When you find a new prime:
At this site we keep database of the 5000 Largest
Known Primes, plus the
top twenty of certain special forms (Cullen, twin,...) If you find
any primes that would fit on this list, please let me know! To see
what is already on the list, you can use the search
Search the list for primes of these forms
Click here for other ways to
search at this site and in this database.