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 181 Chris Howell 1 48.8827 182 KaiMing Poon 3 48.8760 183 Senji Yamashita 11.3333 48.8694 184 Kyle J Johnson 3 48.8629 185 Wolfram Glatte 7 48.8524 186 A.J.Brech 10 48.8522 187 Jiri Jaros 1 48.8448 188 Brian Parsonnet 8 48.8400 189 Brook Harste 2 48.8333 190 Jens Katzur 7 48.8152 191 Axel Amberg 2 48.8021 192 Douglas Grosvenor 16 48.8003 193 Jacques Molné 3 48.7940 194 Mariusz Ługowski 2 48.7904 195 Milan Fňašek 2 48.7858 196 Alex Huetter 2 48.7837 197 Magnus Bergman 1 48.7753 198 Zoltan Kemenes 2 48.7583 199 Kellen Shenton 9 48.7581 200 Ian Dickinson 7 48.7531

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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)).