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 441 Reiner Elgetz 3 47.8425 442 Kimmo Koski 3 47.8378 443 Rick Reynolds 3 47.8373 444 André Ahlfors Dahl 1 47.8283 445 Yuki Yoshigoe 3 47.8229 446 Dustin Waller 2 47.8185 447 Aaron Raimist 3 47.8131 448 Denis Korolev 2 47.8107 449 Yves Gallot 2 47.8013 450 Paweł Dec 3 47.7967 451 James S. Gilliland 3 47.7950 452 Dave Sunderland 3 47.7935 453 Friedrich Yuhma Steinwedel 1 47.7919 454 Michel Johnson 1 47.7877 455 Sam Poulos 2 47.7831 456 Nico Puada 2 47.7830 457 Edward Trice 1 47.7823 458 Toshitaka Kumagai 1 47.7806 459 Mike McDonald 3 47.7710 460 Rob Powell 2.5 47.7700 460 Robert Anthony Lawrence 2.5 47.7700

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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)).