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 81 Andrew M Farrow 4 49.9997 82 James Winskill 3 49.9926 83 Masashi Kumagai 1 49.9772 84 Tim Terry 14 49.9596 85 Lei Zhou 15 49.9286 86 Scott Lee 6 49.9097 87 Tim McArdle 1 49.9091 88 Peyton Hayslette 1 49.8982 89 Jonathan Sipes 16 49.8653 90 Ian Johns 21 49.8383 91 Dennis Sydekum 7 49.8265 92 Benoit Da Mota 5 49.8112 93 Latah Headrick 8 49.8093 94 Frank Doornink 23 49.7952 95 Takahiko Ogawa 25 49.7893 96 Michael Goetz 5 49.7604 97 Derek Gordon 1 49.7454 98 Patrice Salah 1 49.7436 99 Göran Schmidt 50.6667 49.7412 100 Ken Ito 10 49.7376

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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)).