Top persons sorted by score
(Another of the Prime Pages' resources)
The Largest Known Primes Icon
  View this page in:   language help
The Prover-Account Top 20
Persons by: number score normalized score
Programs by: number score normalized score
Projects by: number score normalized 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.

21 Serhiy Gushchak 1 50.4356
22 Sai Yik Tang 32 50.3763
23 David Metcalfe 196 50.0364
24 Masashi Kumagai 2 49.9806
25 Dmitry Domanov 43 49.9132
26 Tim McArdle 1 49.9091
27 Peyton Hayslette 1 49.8982
28 Peter Kaiser 34.3333 49.8826
29 Hiroyuki Okazaki 41 49.8243
30 Derek Gordon 1 49.7454
31 Patrice Salah 1 49.7436
32 Thomas Ritschel 69 49.6794
33 Ronny Willig 41 49.6309
34 Stefan Larsson 35 49.6044
35 Michael Goetz 2 49.5188
36 Tom Greer 134 49.5179
37 Borys Jaworski 20 49.4946
38 Grzegorz Granowski 95 49.4510
39 Ars Technica Team Prime Rib 1 49.2624
40 Michael Curtis 42.5 49.2323

move up list ^
move down list v


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)).