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 411 Michael D. Hittle 3 44.6664 412 Robert Collins 1 44.6575 413 Sergio Cristian 1 44.6535 414 GertJan van Oenen 1 44.6523 415 Radoslaw Szumlakowski 3 44.6487 416 Pekka Pasanen 1 44.6475 417 Philip Englehard 3 44.6379 418 Timothy Sorbera 3 44.6335 419 ChunWei Shen 3 44.6320 420 Reto Keiser 17 44.6298 421 Ross Romer 2 44.6257 422 Joseph Hsieh 1 44.6253 423 Samy Bourghol 3 44.6219 424 Michal Tatárka 2 44.6105 425 Bo Tornberg 9 44.5889 426 Mike Lane 2 44.5813 427 Thierry Fausten 1 44.5811 428 Kevin Woerner 1 44.5797 429 Anders Åström 3 44.5758 430 Zhiyuan Zhan 2 44.5745

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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)).