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 Michael Curtis 46 49.8990 42 Peyton Hayslette 1 49.8982 43 Borys Jaworski 19 49.8344 44 Derek Gordon 1 49.7454 45 Patrice Salah 1 49.7436 46 Honza Cholt 29 49.6170 47 Michael Goetz 4 49.5303 48 Grzegorz Granowski 69 49.3556 49 Ars Technica Team Prime Rib 1 49.2624 50 Ralf Terber 18 49.1632 51 Walter Darimont 2 49.1458 52 Randy Ready 13 49.1180 53 Andrew M Farrow 5 49.0549 54 Ken Ito 12 49.0472 55 Daniel Thonon 78 49.0429 56 Predrag Kurtovic 22 49.0283 57 Sean Humphries 7 49.0192 58 Denis Iakovlev 1 48.9974 59 Martin Vanc 1 48.9657 60 Senji Yamashita 26.3333 48.9435

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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)).