Top persons sorted by score
(Another of the Prime Pages' resources)
The Largest Known Primes Icon
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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.

rankpersonprimesscore
61 Håkan Lind 4 48.5101
62 David Mumper 1 48.4921
63 William de Thomas 4 48.4631
64 David Yost 28 48.4204
65 Seonghwan Kim 22 48.3911
66 Steven Wong 15 48.3883
67 Larry Soule 35.5 48.3673
68 Roman Vogt 2 48.3345
69 Martyn Elvy 1 48.3296
70 Mark Molder 21 48.3147
71 Michael Mamanakis 29 48.2777
72 Frank Schwegler 2 48.2697
73 Sean Humphries 1 48.2609
74 Jan Haller 3 48.2099
75 Predrag Kurtovic 12 48.1883
76 Tsuyoshi Ohsugi 1 48.1767
77 Frank Matillek 5 48.1737
78 Lei Zhou 16 48.1722
79 Mark Rodenkirch 15.3333 48.1716
80 Szymon Banka 4.3333 48.1510

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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)).