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] 59 57.7014 3 Jean Penné's LLR [special, plus, minus] 4721 56.6266 4 Geoffrey Reynolds' srsieve [sieve] 3132 55.9013 5 Pavel Atnashev's PRST [] 220 55.1036 6 EMsieve [sieve] 77 55.0601 7 Anand Nair's CycloSvCUDA sieve [sieve] 41 55.0081 8 David Underbakke's AthGFNSieve [sieve] 1344 54.9752 9 Yves Gallot's Cyclo [special] 51 54.9662 10 Anand Nair's GFNSvCUDA sieve [sieve] 1334 54.9623 11 Yves Gallot's GeneFer [prp, special] 1335 54.9620 12 LLR2 [other] 1088 54.8363 13 Reynolds and Brazier's PSieve [sieve] 2399 54.4922 14 Geoffrey Reynolds' gcwsieve [sieve] 62 54.1065 15 Mikael Klasson's Proth_sieve [sieve] 16 53.6124 16 Phil Carmody's 'K' sieves [sieve] 7 53.6078 16 Paul Jobling's SoBSieve [sieve] 7 53.6078 18 MultiSieve/mtsieve [sieve] 35 52.9336 19 OpenPFGW (a.k.a. PrimeForm) [other, sieve, prp, special, plus, minus, classical] 413 52.7022 20 Yves Gallot's Proth.exe [other, special, plus, minus, classical] 55 51.4814

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)).