Top person sorted by 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 721 Toon Pistorius 1 46.7518 722 Chris Wilson 1 46.7473 723 Juan C. Toledo 1 46.7472 724 Paul York 1 46.7456 725 Andrew R. Booker 1 46.7434 726 MaikeSophie Mittelstädt 1 46.7417 727 Joseph Wilson 1 46.7386 728 Steve Dodd 1 46.7382 729 Matt Smith 1 46.7377 730 Mariusz Szafrański 2 46.7360 731 Javi Jarne 2 46.7359 732 佐藤吉紘 1 46.7349 733 Christian Einvik 1 46.7313 734 Dave Pickles 1 46.7312 735 John R. Bailey 1 46.7284 736 Pete D'Amico 1 46.7247 737 Jonathan Pritchard 1 46.7203 738 Devlin Skahill 1 46.7186 739 Vaclav Koci 1 46.7163 740 William Darney 1 46.7150

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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)).