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

GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

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 81 Magnus Bergman 1 48.7753 82 Scott Gilvey 2 48.6840 83 Vladimir Volynsky 1 48.6527 84 Lei Zhou 17 48.6327 85 Dennis R. Gesker 1 48.6097 86 William de Thomas 4 48.6019 87 Mariusz Szafrański 26 48.5696 88 Håkan Lind 4 48.5101 89 Seiya Tsuji 2 48.5020 90 David Mumper 1 48.4921 91 Jacob Eikelenboom 1 48.4840 92 Vincent Diepeveen 3.5 48.4783 93 Yoshimitsu Kato 1 48.4773 94 Vince Splain 25 48.4399 95 Karsten Freihube 2 48.4271 96 Rick Reynolds 27 48.3987 97 Frank Schwegler 4 48.3957 98 Jiří Bočan 1 48.3935 99 Martyn Elvy 1 48.3296 100 Mark Molder 21 48.3147

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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)).