updates

Prototypes/Prototypes.wl

@@ -31,7 +31,7 @@ Module[ {files},
     "Image.wl", "Language.wl", "Paclet.wl", "Formats.wl",
     "Resources.wl", "Search.wl", "String.wl", "System.wl", "Color.wl",
     "Entities.wl", "WolframAlpha.wl", "Translation.wl","GeoGraphics.wl", "Words.wl", "Packages.wl", "Random.wl",
-    "Cryptography.wl", "Service/What3Words.wl"};
+    "Cryptography.wl", "Service/What3Words.wl", "Primes/Primes.wl"};
     Map[
       Function[ {file},
         AppendTo[ timings, file<>"-Before" -> N@SessionTime[] ];

Prototypes/Source/Primes/Binaries/Linux-x86-64/COPYING

@@ -0,0 +1,24 @@
+BSD 2-Clause License
+
+Copyright (c) 2013 - 2018, Kim Walisch.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+1. Redistributions of source code must retain the above copyright notice, this
+   list of conditions and the following disclaimer.
+2. Redistributions in binary form must reproduce the above copyright notice,
+   this list of conditions and the following disclaimer in the documentation
+   and/or other materials provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
+ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
+ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Prototypes/Source/Primes/Binaries/Linux-x86-64/README.txt

@@ -0,0 +1,69 @@
+primecount 4.4
+May 9, 2018
+Kim Walisch, <[email protected]>
+
+About primecount
+================
+
+  primecount is a command-line program that counts the number of
+  primes below an integer x <= 10^31 using fast implementations of the
+  prime counting function pi(x).
+
+  Homepage: https://github.com/kimwalisch/primecount
+
+Usage examples
+==============
+
+  Open a terminal and run:
+
+  # Count the primes below 10^14
+  ./primecount 1e14
+
+  # Print progress and status information during computation
+  ./primecount 1e20 --status
+
+  # Count primes using Meissel's algorithm
+  ./primecount 2**32 --meissel
+
+  # Find the 10^14th prime using 4 threads
+  ./primecount 1e14 --nthprime --threads=4 --time
+
+Command-line options
+====================
+
+  Usage: primecount x [OPTION]...
+  Count the primes below x <= 10^31 using fast implementations of the
+  combinatorial prime counting function.
+
+  Options:
+
+    -d,    --deleglise_rivat  Count primes using Deleglise-Rivat algorithm
+           --legendre         Count primes using Legendre's formula
+           --lehmer           Count primes using Lehmer's formula
+    -l,    --lmo              Count primes using Lagarias-Miller-Odlyzko
+    -m,    --meissel          Count primes using Meissel's formula
+           --Li               Approximate pi(x) using the logarithmic integral
+           --Li_inverse       Approximate the nth prime using Li^-1(x)
+    -n,    --nthprime         Calculate the nth prime
+    -p,    --primesieve       Count primes using the sieve of Eratosthenes
+           --phi=<a>          phi(x, a) counts the numbers <= x that are
+                              not divisible by any of the first a primes
+           --Ri               Approximate pi(x) using Riemann R
+           --Ri_inverse       Approximate the nth prime using Ri^-1(x)
+    -s[N], --status[=N]       Show computation progress 1%, 2%, 3%, ...
+                              [N] digits after decimal point e.g. N=1, 99.9%
+           --test             Run various correctness tests and exit
+           --time             Print the time elapsed in seconds
+    -t<N>, --threads=<N>      Set the number of threads, 1 <= N <= CPU cores
+    -v,    --version          Print version and license information
+    -h,    --help             Print this help menu
+
+  Advanced Deleglise-Rivat options:
+
+    -a<N>, --alpha=<N>        Tuning factor, 1 <= alpha <= x^(1/6)
+           --P2               Only compute the 2nd partial sieve function
+           --S1               Only compute the ordinary leaves
+           --S2_trivial       Only compute the trivial special leaves
+           --S2_easy          Only compute the easy special leaves
+           --S2_hard          Only compute the hard special leaves
+

Prototypes/Source/Primes/Binaries/MacOSX-x86-64/COPYING

@@ -0,0 +1,24 @@
+BSD 2-Clause License
+
+Copyright (c) 2013 - 2018, Kim Walisch.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+1. Redistributions of source code must retain the above copyright notice, this
+   list of conditions and the following disclaimer.
+2. Redistributions in binary form must reproduce the above copyright notice,
+   this list of conditions and the following disclaimer in the documentation
+   and/or other materials provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
+ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
+ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Prototypes/Source/Primes/Binaries/MacOSX-x86-64/README.txt

@@ -0,0 +1,69 @@
+primecount 4.4
+May 9, 2018
+Kim Walisch, <[email protected]>
+
+About primecount
+================
+
+  primecount is a command-line program that counts the number of
+  primes below an integer x <= 10^31 using fast implementations of the
+  prime counting function pi(x).
+
+  Homepage: https://github.com/kimwalisch/primecount
+
+Usage examples
+==============
+
+  Open a terminal and run:
+
+  # Count the primes below 10^14
+  ./primecount 1e14
+
+  # Print progress and status information during computation
+  ./primecount 1e20 --status
+
+  # Count primes using Meissel's algorithm
+  ./primecount 2**32 --meissel
+
+  # Find the 10^14th prime using 4 threads
+  ./primecount 1e14 --nthprime --threads=4 --time
+
+Command-line options
+====================
+
+  Usage: primecount x [OPTION]...
+  Count the primes below x <= 10^31 using fast implementations of the
+  combinatorial prime counting function.
+
+  Options:
+
+    -d,    --deleglise_rivat  Count primes using Deleglise-Rivat algorithm
+           --legendre         Count primes using Legendre's formula
+           --lehmer           Count primes using Lehmer's formula
+    -l,    --lmo              Count primes using Lagarias-Miller-Odlyzko
+    -m,    --meissel          Count primes using Meissel's formula
+           --Li               Approximate pi(x) using the logarithmic integral
+           --Li_inverse       Approximate the nth prime using Li^-1(x)
+    -n,    --nthprime         Calculate the nth prime
+    -p,    --primesieve       Count primes using the sieve of Eratosthenes
+           --phi=<a>          phi(x, a) counts the numbers <= x that are
+                              not divisible by any of the first a primes
+           --Ri               Approximate pi(x) using Riemann R
+           --Ri_inverse       Approximate the nth prime using Ri^-1(x)
+    -s[N], --status[=N]       Show computation progress 1%, 2%, 3%, ...
+                              [N] digits after decimal point e.g. N=1, 99.9%
+           --test             Run various correctness tests and exit
+           --time             Print the time elapsed in seconds
+    -t<N>, --threads=<N>      Set the number of threads, 1 <= N <= CPU cores
+    -v,    --version          Print version and license information
+    -h,    --help             Print this help menu
+
+  Advanced Deleglise-Rivat options:
+
+    -a<N>, --alpha=<N>        Tuning factor, 1 <= alpha <= x^(1/6)
+           --P2               Only compute the 2nd partial sieve function
+           --S1               Only compute the ordinary leaves
+           --S2_trivial       Only compute the trivial special leaves
+           --S2_easy          Only compute the easy special leaves
+           --S2_hard          Only compute the hard special leaves
+

Prototypes/Source/Primes/Binaries/Windows-x86-64/COPYING

@@ -0,0 +1,24 @@
+BSD 2-Clause License
+
+Copyright (c) 2013 - 2018, Kim Walisch.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+1. Redistributions of source code must retain the above copyright notice, this
+   list of conditions and the following disclaimer.
+2. Redistributions in binary form must reproduce the above copyright notice,
+   this list of conditions and the following disclaimer in the documentation
+   and/or other materials provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
+ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
+ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Prototypes/Source/Primes/Binaries/Windows-x86-64/README.txt

@@ -0,0 +1,68 @@
+primecount 4.4
+May 9, 2018
+Kim Walisch, <[email protected]>
+
+About primecount
+================
+
+  primecount is a command-line program that counts the number of
+  primes below an integer x <= 10^31 using fast implementations of the
+  prime counting function pi(x).
+
+  Homepage: https://github.com/kimwalisch/primecount
+
+Usage examples
+==============
+
+  Open a Command Prompt and run:
+
+  # Count the primes below 10^14
+  > primecount 1e14
+
+  # Print progress and status information during computation
+  > primecount 1e20 --status
+
+  # Count primes using Meissel's algorithm
+  > primecount 2**32 --meissel
+
+  # Find the 10^14th prime using 4 threads
+  > primecount 1e14 --nthprime --threads=4 --time
+
+Command-line options
+====================
+
+  Usage: primecount x [OPTION]...
+  Count the primes below x <= 10^31 using fast implementations of the
+  combinatorial prime counting function.
+
+  Options:
+
+    -d,    --deleglise_rivat  Count primes using Deleglise-Rivat algorithm
+           --legendre         Count primes using Legendre's formula
+           --lehmer           Count primes using Lehmer's formula
+    -l,    --lmo              Count primes using Lagarias-Miller-Odlyzko
+    -m,    --meissel          Count primes using Meissel's formula
+           --Li               Approximate pi(x) using the logarithmic integral
+           --Li_inverse       Approximate the nth prime using Li^-1(x)
+    -n,    --nthprime         Calculate the nth prime
+    -p,    --primesieve       Count primes using the sieve of Eratosthenes
+           --phi=<a>          phi(x, a) counts the numbers <= x that are
+                              not divisible by any of the first a primes
+           --Ri               Approximate pi(x) using Riemann R
+           --Ri_inverse       Approximate the nth prime using Ri^-1(x)
+    -s[N], --status[=N]       Show computation progress 1%, 2%, 3%, ...
+                              [N] digits after decimal point e.g. N=1, 99.9%
+           --test             Run various correctness tests and exit
+           --time             Print the time elapsed in seconds
+    -t<N>, --threads=<N>      Set the number of threads, 1 <= N <= CPU cores
+    -v,    --version          Print version and license information
+    -h,    --help             Print this help menu
+
+  Advanced Deleglise-Rivat options:
+
+    -a<N>, --alpha=<N>        Tuning factor, 1 <= alpha <= x^(1/6)
+           --P2               Only compute the 2nd partial sieve function
+           --S1               Only compute the ordinary leaves
+           --S2_trivial       Only compute the trivial special leaves
+           --S2_easy          Only compute the easy special leaves
+           --S2_hard          Only compute the hard special leaves

Prototypes/Source/Primes/Primes.wl

@@ -0,0 +1,3 @@
+primeCountBinary = FileNameJoin[{DirectoryName[$InputFileName], "Binaries", $SystemID, "primecount"}] ];
+
+PrimeCount[n_Integer] := ToExpression[ StringTrim[ RunProcess[ {primeCountBinary, ToString[n]}, "StandardOutput"]]];

Prototypes/Source/Usage.wl

@@ -140,3 +140,6 @@ What3Words::usage = "What3Words[location] and What3Words[words] use the What3Wor
 
 (* Crypto *)
 $HashTypes::usage = "$HashTypes gives the list of supported hash types";
+
+(* Primes *)
+PrimeCount::usage = "PrimeCount[n] gives the number of primes less than or equal to 'n'. Uses binaries from: https://github.com/kimwalisch/primecount";