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 461 Seab Herron 2 47.7164 462 José Miguel 2 47.7139 463 Richard H Taylor 2 47.7137 464 Miroslav Kupka 2 47.7076 465 Tom Kledzik 2 47.7039 466 Marian Brockerhoff 1 47.6942 467 Charlotte Woodrow 2 47.6885 468 Uwe Pilz 2 47.6882 469 Thomas Kopp 2 47.6862 470 Florian Leudesdorff 2 47.6856 471 Sebastien Deram 2 47.6821 472 Alessandro Puppi 2 47.6803 473 Don Wellck 2 47.6802 474 Tapio Rajala 1 47.6789 475 Sam Black 2 47.6770 476 Sandor Hasznos 2 47.6748 477 José Andrade 2 47.6731 478 Uli Abromeit 2 47.6689 479 Dr. Karsten Steffens 2 47.6634 480 Ruediger K. Eckhard 3 47.6621

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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)).