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 person primes score 41 Borys Jaworski 20 50.2155 42 David Metcalfe 150 49.9848 43 Masashi Kumagai 2 49.9806 44 Michael Curtis 49 49.9533 45 Dmitry Domanov 43 49.9223 46 Tim McArdle 1 49.9091 47 Peyton Hayslette 1 49.8982 48 Honza Cholt 31 49.7538 49 Derek Gordon 1 49.7454 50 Patrice Salah 1 49.7436 51 SRBase 59.5 49.6518 52 Michael Goetz 4 49.5303 53 Ken Ito 10 49.4774 54 Grzegorz Granowski 68 49.3507 55 Ars Technica Team Prime Rib 1 49.2624 56 Jordan Romaidis 8 49.2582 57 Rod Skinner 4 49.2391 58 Ralf Terber 20 49.2290 59 Predrag Kurtovic 26 49.2077 60 Daniel Thonon 83 49.2053

move up list

move down list

Notes:

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)).