I’ve been searching for primes of the form k*2^n-1 with the 15k project for a while now and became interested in finding candidates, k, that would produce a top-5000 prime the fastest. How long it takes to find a top-5000 prime for a particular k value depends on many factors: how efficient sieving is, how long LLR takes for that particular k, and how often that k produces primes. To estimate this, I started with k’s formed by all combinations of the first 11 primes and k>1000. For each k, measure

• The number of candidates left after sieving n=0-10000 to 100M

• The number of primes in the range n=0-10000

• The number of bits in k (the time for LLR is proportional to this)

The result of this can be summed up in a couple of lists. First is the top-20 list of the k’s that had the most primes in the range n=0-10000. All of the k’s here are divisible by 15k.

Code:

# Primes K
81 8331405
77 944876594805
76 7526103585
73 743411955
72 685084785
71 640049865
71 41237498445
69 1818438765
69 169215941895
69 152125131763605
68 49335
68 965426385
68 1272300315
67 285413005185
67 1212935295
67 553437885
67 51449055
67 377481716535
66 578931045
66 3009765

Next is the top-20 list for the k’s that are estimated to be fastest to find a top-5000 prime with. The score is related to how long it should take to find a top-5000 prime in the range of k*2^200000-1. Thus, the smaller the better. All but 2 of the top 20 are divisible by 15.

Code:

Score K Primes already on top-5000 list
2481891 8331405 None!
2545731 49335 184364, 199133, 242161
2552143 26565 198349, 217001
2744279 25935 None!
2819593 6555 None!
2850866 19635 189197
2955186 373065 None!
2990424 102765 None!
3013362 62985 None!
3062642 67773 None!
3085355 5865 None!
3098476 3009765 None!
3119285 7526103585 None!
3123287 743411955 None!
3180954 2330445 None!
3197466 19437 None!
3222459 465465 None!
3238416 2805 185593, 192628, 200027, 200212, 203574, 212227, 230666
3242361 685084785 None!
3244999 2667885 None!

I tried out the first K, 8331405, and found a top-5000 prime within 24 hours!