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 124 Sai Yik Tang 20 46.3157 125 Oscar Tutusaus Santandreu 10 46.3073 126 Rodger M. Ewing 5 46.3065 127 Gordon Spence 1 46.2958 128 Andreas Adolfsson 9 46.2944 129 Bryan Koen 2 46.2743 130 John Cosgrave 4 46.2605 131 Simeon I. B. Ayeni 4 46.2577 132 Michael Reifschneider 13 46.2533 133 Sascha Beat Dinkel 16 46.2432 134 Steven Wong 8 46.2390 135 Lars Dausch 5 46.2134 136 John Jacobs 2 46.2019 137 Seiya Tsuji 1 46.1963 138 Ryan English 2 46.1926 139 Andrei Ryjkov 14 46.1899 140 Gerald C. Snyder 2 46.1836 141 Dale Laluk 15 46.1789 142 Jeff Anderson-Lee 8 46.1768 143 Scott Yoshimura 1 46.1681

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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)).