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 142 Jeff Anderson-Lee 8 46.1768 143 Scott Yoshimura 1 46.1681 144 John S Morris III 11 46.1668 145 HansJürgen Bergelt 17 46.1647 146 Peter Grobstich 9 46.1645 147 Marshall Bishop 3 46.1532 148 Alexander Gramolin 6 46.1475 149 Takashi Iwasaki 11 46.1180 150 Bartlett Henderson 12 46.1139 151 Takeshi Watanabe 10 46.0986 152 Michael Angel 10.6667 46.0907 153 Wolfgang Schwieger 10 46.0789 154 Kimmo Myllyvirta 10 46.0601 155 Ronny Willig 3 46.0552 156 Bruno Courty 10 46.0520 157 Joseph Bohanon 3 46.0271 158 René Dohmen 5 46.0148 159 Yves Gallot 3 46.0006 160 Chris Siegert 13 45.9760 161 Ken Micom 3 45.9661

move up list

move down list

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)).