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 161 Uwe Hass 6 49.3161 162 JanPeter Fischer 13 49.3069 163 Jens Katzur 11 49.3014 164 Matt Jurach 9 49.2930 165 James Schumacher 9 49.2892 166 Mariusz Ługowski 3 49.2873 167 Lorin Arnold 6 49.2862 168 Howard Shi 5 49.2740 169 Ars Technica Team Prime Rib 1 49.2624 170 Florian Baur 3 49.2518 171 Rod Skinner 4 49.2391 172 Vincent Diepeveen 5 49.2227 173 A.J.Brech 13 49.2218 174 Frank Meador 5 49.2169 175 Liam McGonegal 7 49.2133 176 William Byerly 8 49.2100 177 Anthony Ayiomamitis 4 49.1905 178 Alvaro Morera 8 49.1899 179 Eric Nietering 11 49.1864 180 Robert L. Clark 10 49.1854

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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)).