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 221 Rico Gefreiter 8 49.0877 222 Dao Heng Liu 10 49.0798 223 Rafael Trigueiro 4 49.0735 224 Fangping Wen 7 49.0697 225 Daniel Wimmer 3 49.0586 226 Eudy Silva 5 49.0478 227 Ian Dickinson 9 49.0476 228 Steven Wong 9 49.0434 229 Keith Reinhardt 11 49.0397 230 Eric Nietering 8 49.0289 231 Gary Bauer 7 49.0263 232 A.J.Brech 9 49.0206 233 Christian Guilleminot 8 49.0203 234 William Bailey 2 49.0108 235 Frederik Schiøler 7 49.0072 236 Denis Iakovlev 1 48.9974 237 John S Chambers 3 48.9893 238 William Pomeranke 2 48.9834 239 Km Kim 2 48.9817 240 Sean Humphries 2 48.9729

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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)).