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 61 Martin Vanc 1 48.9657 62 Senji Yamashita 26.3333 48.9435 63 Michael Herder 1 48.9335 64 Sturle Sunde 1 48.9324 65 Jochen Beck 11 48.9143 66 Nayan Hajratwala 1 48.9067 67 Jordan Romaidis 7 48.8944 68 Brook Harste 7 48.8875 69 Greg Miller 14 48.8433 70 Alen Kecic 10 48.8351 71 Frank Meador 1 48.8338 72 Bill Cavnaugh 37 48.7885 73 Magnus Bergman 1 48.7753 74 Ricky L Hubbard 19 48.7444 75 Steven Wong 27 48.7053 76 Scott Gilvey 2 48.6840 77 A.J.Brech 52 48.6580 78 Vladimir Volynsky 1 48.6527 79 Dennis R. Gesker 1 48.6097 80 William de Thomas 4 48.6019

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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)).