1PRIMECOUNT(1) PRIMECOUNT(1)
2
6 primecount - count prime numbers
7
9 primecount x [options]
10
12 Count the number of primes less than or equal to x (<= 10^31) using
13 fast implementations of the combinatorial prime counting function
14 algorithms. By default primecount counts primes using Xavier Gourdon’s
15 algorithm which has a runtime complexity of O(x^(2/3) / log^2 x)
16 operations and uses O(x^(2/3) * log^3 x) memory. primecount is
17 multi-threaded, it uses all available CPU cores by default.
18
20 -d, --deleglise-rivat
21 Count primes using the Deleglise-Rivat algorithm.
22 -g, --gourdon
24 Count primes using Xavier Gourdon’s algorithm (default algorithm).
25 -l, --legendre
27 Count primes using Legendre’s formula.
28 --lehmer
30 Count primes using Lehmer’s formula.
31 --lmo
33 Count primes using the Lagarias-Miller-Odlyzko algorithm.
34 -m, --meissel
36 Count primes using Meissel’s formula.
37 --Li
39 Approximate pi(x) using the logarithmic integral.
40 --Li-inverse
42 Approximate the nth prime using Li^-1(x).
43 -n, --nth-prime
45 Calculate the nth prime.
46 -p, --primesieve
48 Count primes using the sieve of Eratosthenes.
49 --phi X A
51 phi(x, a) counts the numbers <= x that are not divisible by any of
52 the first a primes.
53 --Ri
55 Approximate pi(x) using the Riemann R function.
56 --Ri-inverse
58 Approximate the nth prime using Ri^-1(x).
59 -s, --status[=NUM]
61 Show the computation progress e.g. 1%, 2%, 3%, ... Show NUM digits
62 after the decimal point: --status=1 prints 99.9%.
63 --test
65 Run various correctness tests and exit.
66 --time
68 Print the time elapsed in seconds.
69 -t, --threads=NUM
71 Set the number of threads, 1 <= NUM <= CPU cores. By default
72 primecount uses all available CPU cores.
73 -v, --version
75 Print version and license information.
76 -h, --help
78 Print this help menu.
79
81 --P2
82 Compute the 2nd partial sieve function.
83 --S1
85 Compute the ordinary leaves.
86 --S2-trivial
88 Compute the trivial special leaves.
89 --S2-easy
91 Compute the easy special leaves.
92 --S2-hard
94 Compute the hard special leaves.
95 Tuning factor
97 The alpha tuning factor mainly balances the computation of the S2_easy
98 and S2_hard formulas. By increasing alpha the runtime of the S2_hard
99 formula will usually decrease but the runtime of the S2_easy formula
100 will increase. For large pi(x) computations with x >= 10^25 you can
101 usually achieve a significant speedup by increasing alpha.
102 The alpha tuning factor is also very useful for verifying pi(x)
104 computations. You compute pi(x) twice but for the second computation
105 you use a slightly different alpha factor. If the results of both pi(x)
106 computations match then pi(x) has been verified successfully.
107 -a, --alpha=NUM
109 Set the alpha tuning factor: y = x^(1/3) * alpha, 1 <= alpha <=
110 x^(1/6).
111
113 --AC
114 Compute the A + C formulas.
115 --B
117 Compute the B formula.
118 --D
120 Compute the D formula.
121 --Phi0
123 Compute the Phi0 formula.
124 --Sigma
126 Compute the 7 Sigma formulas.
127 Tuning factors
129 The alpha_y and alpha_z tuning factors mainly balance the computation
130 of the A, B, C and D formulas. When alpha_y is decreased but alpha_z is
131 increased then the runtime of the B formula will increase but the
132 runtime of the A, C and D formulas will decrease. For large pi(x)
133 computations with x >= 10^25 you can usually achieve a significant
134 speedup by decreasing alpha_y and increasing alpha_z. For convenience
135 when you increase alpha_z using --alpha-z=NUM then alpha_y is
136 automatically decreased.
137 Both the alpha_y and alpha_z tuning factors are also very useful for
139 verifying pi(x) computations. You compute pi(x) twice but for the
140 second computation you use a slightly different alpha_y or alpha_z
141 factor. If the results of both pi(x) computations match then pi(x) has
142 been verified successfully.
143 --alpha-y=NUM
145 Set the alpha_y tuning factor: y = x^(1/3) * alpha_y, 1 <= alpha_y
146 <= x^(1/6).
147 --alpha-z=NUM
149 Set the alpha_z tuning factor: z = y * alpha_z, 1 <= alpha_z <=
150 x^(1/6).
151
153 primecount 1000
154 Count the primes <= 1000.
155 primecount 1e17 --status
157 Count the primes <= 10^17 and print status information.
158 primecount 1e15 --threads 1 --time
160 Count the primes <= 10^15 using a single thread and print the time
161 elapsed.
162
164 https://github.com/kimwalisch/primecount
165
167 Kim Walisch <kim.walisch@gmail.com>
168 07/21/2023 PRIMECOUNT(1)