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} prothsearch.net for the username and password from a valid email address.List of primes for k < 300List of primes for 300 < k < 600 |

Primes k.2 – 1 ^{n} |
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*.2^{n}*+*1 with 2^{n}* > 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!

- Download Program from Chris K. Caldwell’s Prime Site.
**Links to tables of ranges**

Primes of the form *k*.2^{n}+1 are interesting as possible factors of **Fermat numbers** *F*_{m} = 2^{r} + 1, where *r* = 2^{m} 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*.2^{n}+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.2^{n}+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.2^{n}+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 |