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 801 Thomaz Santana 1 46.6203 802 Steve Hawker 1 46.6085 803 Holger Venjakob 1 46.6004 804 John T Marshall 1 46.6003 805 Ryan Dark 1 46.5983 806 Dongyan Jia 1 46.5935 807 Jace Tyndall 1 46.5865 808 CHEUNG Tak Chuen 1 46.5856 809 Johan Hefer 1 46.5842 810 Alex_soldier 1 46.5810 811 Jaroslav Čampulka 1 46.5662 812 Rick Smith 1 46.5614 813 Detlef Lexut 1 46.5543 814 Tim Ranch 1 46.5517 815 Kenneth Collins Freeman 1 46.5500 816 Richard Cervelle 1 46.5465 817 Philipp Schulz 1 46.5238 818 Ian Keogh 1 46.5194 819 Warach Veeranonchai 1 46.5114 820 Jim Wilkins 1 46.4658

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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)).