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 941 Nuutti Kuosa 1.3333 40.2057 942 Darren Smith 2 40.1303 943 Andreas Enge 4.5 40.1249 944 Greg Childers 18 39.9749 945 Mike Oakes 18.75 39.5882 946 Bo Tornberg 2 39.5710 947 Keith Klahn 1 39.5652 948 Phil Carmody 0.7833 39.5625 949 Leon Marchal 0.3333 39.5549 950 Bouk de Water 38.6999 39.1840 951 Nicholas M. Glover 0.5 39.1453 952 Antal Járai 1.2 38.9168 952 Gabor Farkas 1.2 38.9168 954 S. Urushihata 2 38.6840 955 Zoltán Járai 0.8 38.5025 955 Timea Csajbok 0.8 38.5025 955 János Kasza 0.8 38.5025 958 Maia Karpovich 3 38.3286 959 Gábor E. Gévay 0.4 37.8356 959 Magyar Péter 0.4 37.8356 959 Szekeres Béla 0.4 37.8356

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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)).