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 341 Fred Roberts 3 48.4333 342 Håkan Lind 2 48.4163 343 Sizhao Deng 4 48.4022 344 Seiya Tsuji 1 48.3970 345 Jiří Bočan 1 48.3935 346 Mario Lein 4 48.3923 347 Alain Dumange 4 48.3800 348 Adrian Schori 6 48.3753 349 José Andrade 3 48.3652 350 Teppo Tohmola 5 48.3616 351 Kenneth J. Brazier 4 48.3587 352 Senji Yamashita 2 48.3513 353 Dirk Sellsted 3 48.3351 354 Tomasz Kaczmarek 4 48.3307 355 Martyn Elvy 1 48.3296 356 Gregory Bennett 4 48.3103 357 Wolfgang Schmidt 3 48.3055 358 Alan StPierre 4 48.2974 359 Jörg Meili 4 48.2961 360 Howard Gordon 1 48.2938

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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)).