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 681 Jiří Chovanec 1 46.7891 682 Eduard Cerny 1 46.7880 683 Kotaro Osada 1 46.7868 684 Naoki Yoshioka 1 46.7813 685 Jerry Lin 1 46.7809 686 Michael Russell 1 46.7801 687 Masato Furushima 1 46.7778 688 Xinle Song 1 46.7752 689 Michael Schoeberl 1 46.7713 690 Stefan Lichtenwimmer 1 46.7682 691 Keika Hatanaka 1 46.7680 692 Amy Chambers 1 46.7646 693 Guillaume Laurent 1 46.7586 694 Joerg Steinmetz 1 46.7585 695 Valentin Noxe 1 46.7569 696 John Fleischman 1 46.7550 697 Ladislav Koudelka 1 46.7546 698 Josh Closs 1 46.7538 699 Tod Slakans 2 46.7534 700 Toon Pistorius 1 46.7518

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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)).