Top program sorted by score

At this site we keep several lists of primes, most notably the list of the 5,000 largest known primes. Who found the most of these record primes? We keep separate counts for persons, projects and programs. To see these lists click on 'number' to the right.

Clearly one 100,000,000 digit prime is much harder to discover than quite a few 100,000 digit primes. Based on the usual estimates we score the top persons, provers and projects by adding ‎(log n)³ log log n‎ for each of their primes n. Click on 'score' to see these lists.

Finally, to make sense of the score values, we normalize them by dividing by the current score of the 5000th prime. See these by clicking on 'normalized score' in the table on the right.

rank program primes score 1 Mihai Preda's GpuOwl [prp, special] 1 58.0015 2 George Woltman's Prime95 [special] 56 57.7014 3 Jean Penné's LLR [special, plus, minus] 4662 56.7255 4 Geoffrey Reynolds' srsieve [sieve] 2719 56.0009 5 David Underbakke's AthGFNSieve [sieve] 1814 55.3272 6 Pavel Atnashev's PRST [] 318 55.3185 7 Yves Gallot's GeneFer [prp, special] 1806 55.3182 8 Anand Nair's GFNSvCUDA sieve [sieve] 1804 55.3182 9 EMsieve [sieve] 102 55.0599 10 Anand Nair's CycloSvCUDA sieve [sieve] 40 55.0081 11 LLR2 [other] 1093 54.9968 12 Yves Gallot's Cyclo [special] 47 54.9660 13 Reynolds and Brazier's PSieve [sieve] 2045 54.5273 14 Geoffrey Reynolds' gcwsieve [sieve] 56 54.3431 15 MultiSieve/mtsieve [sieve] 40 53.6242 16 Mikael Klasson's Proth_sieve [sieve] 13 53.6116 17 Phil Carmody's 'K' sieves [sieve] 7 53.6078 17 Paul Jobling's SoBSieve [sieve] 7 53.6078 19 OpenPFGW (a.k.a. PrimeForm) [other, sieve, prp, special, plus, minus, classical] 571 52.7465 20 Yves Gallot's Proth.exe [other, special, plus, minus, classical] 51 51.4929

Notes:

The list above show the programs that are used the most (either by number or score). In some ways this is useless because we are often comparing apples and oranges, that is why the comments in brackets attempt to say what each program does. See the help page for some explanation of these vague categories

Score for Primes

To find the score for a person, program or project's primes, we give each prime n the score (log n)³ log log n; and then find the sum of the scores of their primes. For persons (and for projects), if three go together to find the prime, each gets one-third of the score. Finally we take the log of the resulting sum to narrow the range of the resulting scores. (Throughout this page log is the natural logarithm.)

How did we settle on (log n)³ log log n? For most of the primes on the list the primality testing algorithms take roughly O(log(n)) steps where the steps each take a set number of multiplications. FFT multiplications take about

O( log n ^. log log n ^. log log log n )

operations. However, for practical purposes the O(log log log n) is a constant for this range number (it is the precision of numbers used during the FFT, 64 bits suffices for numbers under about 2,000,000 digits).

Next, by the prime number theorem, the number of integers we must test before finding a prime the size of n is O(log n) (only the constant is effected by prescreening using trial division).  So to get a rough estimate of the amount of time to find a prime the size of n, we just multiply these together and we get

O( (log n)³ log log n ).

Finally, for convenience when we add these scores, we take the log of the result.  This is because log n is roughly 2.3 times the number of digits in the prime n, so (log n)³ is quite large for many of the primes on the list. (The number of decimal digits in n is floor((log n)/(log 10)+1)).