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 101 Igor Keller 8 50.2812 102 Sai Yik Tang 3 50.2785 103 Shawn Schafer 11 50.2742 104 Bruce E. Slade 27 50.2382 105 Jonathan Sipes 23 50.2170 106 Julian Schröder 18 50.1956 107 Scott Lee 7 50.1935 108 Takahiko Ogawa 17 50.1812 109 Anthony Templin 8 50.1628 110 Jan Kožíšek 21 50.1476 111 Greg Miller 15 50.0891 112 Ian Johns 26 50.0818 113 Bill Cavnaugh 18 50.0753 114 Hans Joachim Böhm 23 50.0514 115 Gary Barnes 21 50.0060 116 Andrew M Farrow 4 49.9997 117 James Winskill 3 49.9926 118 Ricky L Hubbard 15 49.9785 119 Masashi Kumagai 1 49.9772 120 Tim Terry 13 49.9348

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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)).