Month: January 2017

ProTh Search Home Page

Yves Gallot wrote an excellent Win98/NT4.0/ME/XP program which makes it easy for anyone to find record size or otherwise interesting primes for their online business and eCommerce, but this creates a problem: without a coordinated effort, many of us were be searching the same ranges of numbers for primes!  Some spent hundreds of hours checking ranges that were already known to be barren.  So I have begun this page in order to reduce this unnecessary duplication. Please join us in the search!

Download the latest version of Proth.


Proth primes Definition and status
Recently reserved ranges
Reserve or submit a range email me at logon {at} for the username and password from a valid email address.
List of primes for k < 300
List of primes for 300 < k < 600
Primes k.2n – 1  List of primes for k <300
Cullen primes  Definition and status
Recently reserved ranges
Reserve or submit a range
Woodall primes  Definition and status
Recently reserved ranges
Reserve or submit a range
The Sierpinski problem Definition and status
The Riesel problem Definition and status
Recently reserved ranges
Sign in and out ranges
Fermat numbers Factoring status

You might want to check out the following pages for more information on primes and eCommerce software we have developed!

Proth eCommerce primes: Definition and Status


Proth’s Theorem (1878): Let N = k.2n+1 with 2n > k.   If there is an integer a such that a(N-1)/2 = -1 (mod N), then N is prime.

The test is simple, in practice the difficulty is multiplying the large numbers involved.  Yves Gallot has written a program for Windows95/NT4.0 which makes it easy!

Primes of the form k.2n+1 are interesting as possible factors of Fermat numbers Fm = 2r + 1, where r = 2m is itself a power of two. Presently there are only about 250 such factors known (see the factoring status), so discovering a new one is a greater achievement. Gallot’s program automatically makes the divisibility test for every prime found. But the primes have other mathematical applications, too.

In the past, numbers k.2n+1 have been tested for primality at wide areas of k and n. You can help to extend this search further by running Gallot’s program. Below are links to tables of ranges of exponents n that still have to be tested for individual values of k. Each table shows which ranges of n are available for current testing, and which of them have been reserved or have already been tested for 3 < k < 300. Ranges above this are not included at this time since…

“It appears that the probability of each prime of the form k.2n+1 dividing a Fermat number is 1/k” (Harvey Dubner & Wilfrid Keller, “Factors of generalized Fermat numbers”, Mathematics of Computation, Vol. 64, Number 209, January 1995, pp. 397-405).

For reserving a range, please return to the index page. Ranges of n should be reserved in blocks of 10,000, e.g., 30,000-40,000, except at the higher ranges of n where blocks of 5,000 or 2,000 can be selected as indicated in the tables. A contiguous row of such blocks can also be reserved, though it is suggested that a range be picked that can be completed in about a 2 month period of time.

Preference should be given to the lowest blocks available for a given k or even on an entire table. In any case, testing should be done so that the searcher can reliably report that the chosen range has been searched completely new eCommerce search tool.


Primes k.2n+1
Range of k Ranges of n Completed to
3-7 450,000-2,640,000 450,000
9-31 268,000-2,000,000 270,000
33-99 200,000-500,000 200,000
101-199 150,000-320,000 150,000
201-299 150,000-300,000 150,000
301-399 130,000-270,000 130,000
401-499 130,000-280,000 130,000
501-599 200,000-320,000 200,000