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 361 Davide Bressani 2 47.9436 362 James S. Gilliland 4 47.9422 363 Peter Riesen 1 47.9390 364 Andy Brady 1 47.9373 365 Nicholas Mukanos 3 47.9372 366 Rob Powell 3.5 47.9364 366 Robert Anthony Lawrence 3.5 47.9364 368 Damian Szeluga 0.3333 47.9359 368 Bartosz Kupidura 0.3333 47.9359 370 Mike Kinney 4 47.9232 371 Randy Sundquist 1 47.9205 372 Robert Lacroix 3 47.9178 373 Jason Jung 3 47.9140 374 Guo Hua Miao 2 47.9130 375 Robert Michael Andrews 3 47.9126 376 Nicholas Mohacsy 2 47.9088 377 Lukasz Piotrowski 3 47.8935 378 Vadim Bulanov 3 47.8865 379 Hendrik Schawe 4 47.8864 380 Sean Matheis 3 47.8839

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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)).