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 181 William Byerly 7 49.1824 182 Lukasz Piotrowski 4 49.1729 183 Julian Schröder 9 49.1697 184 Walter Darimont 2 49.1458 185 David Hua 4 49.1448 186 Brandon Wharton 10 49.1411 187 Darren Li 5 49.1393 188 Keith Reinhardt 13 49.1384 189 Randy Ready 8 49.1258 190 Paul Underwood 46.8333 49.1193 191 Daniel Thonon 14 49.1188 192 Brian Parsonnet 10 49.1060 193 Eudy Silva 6 49.1029 194 Scott Earle 3 49.0986 195 Dao Heng Liu 10 49.0798 196 Rafael Trigueiro 4 49.0735 197 Vladislav Ketamino 9 49.0564 198 Bruce Marler 9 49.0221 199 Jonathan Gehrke 9 49.0019 200 Takeshi Nakamura 10 48.9994

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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)).