Top persons sorted by score
(Another of the Prime Pages' resources)
The Largest Known Primes Icon
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GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

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 Peter Benson 235 50.8999
22 Konstantin Agafonov 1 50.8197
23 Tom Greer 66 50.8002
24 Hiroyuki Okazaki 30 50.6492
25 Michael Schulz 1 50.5434
26 Karsten Klopffleisch 1 50.5009
27 Ronny Willig 156 50.4929
28 Wolfgang Schwieger 32 50.4859
29 Serhiy Gushchak 1 50.4356
30 Stefan Larsson 52 50.4110
31 Ed Goforth 15 50.3964
32 Yair Givoni 1 50.3617
33 Sai Yik Tang 15 50.3508
34 Cesare Marini 1 50.3029
35 Thomas Ritschel 93 50.2473
36 Peter Kaiser 40.3333 50.0227
37 David Metcalfe 158 49.9919
38 Masashi Kumagai 2 49.9806
39 Dmitry Domanov 46 49.9299
40 Tim McArdle 1 49.9091

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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)).