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 project primes score 1 Great Internet Mersenne Prime Search by Woltman & Kurowski 13 55.8432 2 Seventeen or Bust 11 51.3943 3 PrimeGrid 1018 50.0982 4 Riesel Prime Search 956 48.9000 5 Riesel Sieve Project 31 48.3724 6 No Prime Left Behind (formerly: PrimeSearch) 966 48.3227 7 The Prime Sierpinski Problem 12 48.2312 8 321search 8.5 47.4915 9 Yves Gallot's GFN Search Project 30 47.2487 10 Free-DC's Prime Search 330 46.7246 11 Prime Internet Eisenstein Search 160 46.6993 12 GFN 2^17 Sieving project 2.5 46.6601 13 12121 Search 15 46.5517 14 Conjectures 'R Us 33 45.8530 15 Mat's Prime Search 4 45.3451 16 15k*2^n-1 search 36 45.1604 17 GFN 2^16 Sieving project 3 45.1137 18 Riesel Base 5 31 45.0522 19 Generalized Woodall Prime Search 21 44.0824 20 GFN 2^15 Sieving project 5.5 43.5407

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)).