Primoproth Co-ordinated Search

Primoproths are prime numbers of the form p#.2^n +/- 1 (^ = to the power of) and were named as such by Henri Lifchitz. More can be found on Henri's homepage

The primorial, designated by the # symbol, is mathematical shorthand for the product of the primes up given prime number, p.

For example 5# = 2*3*5 = 30. If n=1 then 5#.2^1 - 1 = 59 is a prime, and 5#.2^1+1 = 61 is also prime. 59 and 61 are both primoproths. But so is 23#.2^311147-1, and this is a number which is 93,673 digits long. By way of comparison the total number of atoms in the universe is probably a number which is less than 80 digits long.

Primoproth series, of the form p#.2^n +/- 1, with p a prime and constant, and n variable, will probably contain a greater density of primes than a similar series for a non primorial p. This is because no number in the series will contain factors smaller than p. As an example, the series 67#.2^n+1 contains 100 primes in the first 60,000 n, more than might be expected for an average series.

It might be expected that there will be more prime "specials" than for similar series where k is non primorial, because of this higher density of primes. The third highest Cunningham Chain of the second kind, length 2, 13#.2^23871 is of this form. Records found by this author, or others, are shown here.

This site concentrates on Primoproths in the range 11# to 509#. Smaller values of p# (3, 5, and 7) are caught by other co-ordinated searches, under ranges for 3, 15, and 105. The results of these, as shown in the primes section are taken from The search for k.2^n+1 and A search for k.2^n-1

I would recommend taking a block of 10,000 for testing primality, more if you are running a prime sieve. To reserve a block simply email me on giving me details of your name, the p# (+ or -) reserved, and the range reserved. When you are complete, email the output from prime proving software and I will credit you with the find. If you find a prime large enough to get on the list of large primes held by Chris Caldwell, then go ahead and list it!! At the moment it is necessary to adjust the result format to match to the submissable form k.2^n+/-1. The full numeric values for k=p#/2 are shown here. The n value should be adjusted down by 1. I will endeavour to update the reserved ranges once a week, in order to avoid duplication.

If you need more user friendly output for further analysis, then I can send you information in an Excel spreadsheet. If information from this site is used in any research then full acknowledgement is a must, and please email me any paper referencing this site. I plan to place technical resources here in time.

Go to software details

Go to theoretical contributions

Go to reserved ranges for p#.2^n-1

Go to reserved ranges for p#.2^n+1

Primes found to date p#.2^n-1

Primes found to date p#.2^n+1

Records to date


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Address questions about this web page to: Robert Smith

Last updated 19 October 2002