Top persons sorted by score (Another of the Prime Pages' resources)

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.

rank person primes score 41 Michael Goetz 3 49.5234 42 Michael Curtis 43.5 49.4511 43 Grzegorz Granowski 76 49.3850 44 Rob Gahan 11 49.3108 45 Ars Technica Team Prime Rib 1 49.2624 46 Walter Darimont 2 49.1458 47 Randy Ready 13 49.1180 48 Ralf Terber 12 49.0479 49 Sean Humphries 7 49.0192 50 Denis Iakovlev 1 48.9974 51 Senji Yamashita 31.3333 48.9765 52 Martin Vanc 1 48.9657 53 Michael Herder 1 48.9335 54 Sturle Sunde 1 48.9324 55 Nayan Hajratwala 1 48.9067 56 Brook Harste 9 48.8932 57 Frank Meador 2 48.8415 58 Daniel Thonon 63 48.8184 59 Magnus Bergman 1 48.7753 60 Alen Kecic 8 48.6928

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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)).