Top persons sorted by score (Another of the Prime Pages' resources)

The Prover-Account Top 20
Persons by: number score normalized score
Programs by: number score normalized score
Projects by: number score normalized 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 610 Tapio Rajala 4 43.9205 611 Krzysztof Kozłowski 1 43.9135 612 Dave Sanny 1 43.9098 613 Michael Utiuzh 1 43.9061 614 Minki Li 1 43.9052 615 Harsh Aggarwal 1 43.9024 616 Alexey Derkach 1 43.9017 617 Brian Carpenter 1 43.8852 618 Ovidiu Cioca 1 43.8810 619 Rickie Marsh 1 43.8794 620 Ryo Takeda 1 43.8747 621 Jan Fiedler 1 43.8696 622 Kevin Lannigan 1 43.8690 623 Janne Husu 1 43.8679 624 Paul Chism 1 43.8649 625 Nikolay Dovgiy 1 43.8630 626 Jun Ying 1 43.8547 627 Tony Gott 2 43.8521 628 Brian J Dahlman 2 43.8521 629 Peter Coels 1 43.8392

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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)).