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 21 Randall Scalise 262 50.7753 22 Hiroyuki Okazaki 42 50.5763 23 Michael Schulz 1 50.5434 24 Karsten Klopffleisch 1 50.5009 25 Serhiy Gushchak 1 50.4356 26 Sai Yik Tang 22 50.3616 27 Wolfgang Schwieger 26 50.3575 28 Cesare Marini 1 50.3029 29 Ronny Willig 142 50.1891 30 Stefan Larsson 43 50.0646 31 David Metcalfe 180 50.0152 32 Masashi Kumagai 2 49.9806 33 Dmitry Domanov 42 49.9111 34 Tim McArdle 1 49.9091 35 Peyton Hayslette 1 49.8982 36 Michael Curtis 44.5 49.8868 37 Peter Kaiser 33.3333 49.8803 38 Borys Jaworski 22 49.8426 39 Derek Gordon 1 49.7454 40 Patrice Salah 1 49.7436

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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)).