gaussianprimes 1.0.1
Generate Gaussian primes.
License: MIT
Requires Python: >=3.7

# gaussian-prime-sieve

> A program to generate primes in the Gaussian integers with the Sieve of Eratosthenes.

## Table of Contents

- [Gaussian Integers](#gaussian-integers)
- [Install](#install)
- [Command line usage](#command-line-usage)
- [Python API](#python-api)
- [Algorithm](#algorithm)
- [C++ Implementation](#c-implementation)
- [Applications](#applications)
- [Tests](#tests)
- [License](#license)

## Gaussian Integers

The **Gaussian integers** are complex numbers of the form a + bi where a and b are integers and i is the **imaginary unit**. While the Gaussian integers are a two dimensional extension of the **rational integers** (the usual integers we know and love), they possess nearly all of the same essential properties. Gaussian integers can be factored and the notion of primality is well-defined. The **norm** of a Gaussian integer gives a measure of its size; the analogy in the rational integers is absolute value. Equipped with these tools, sieve methods can be used to compute primes in this extension of our usual integer system.

The goal of this project is to build an efficient C++ implementation of a prime generating sieve in the Gaussian integers and to provide a library for deeper exploration. A natural starting point is the _Sieve of Eratosthenes_, which generates primes in the rational integers. The 2-dimensional nature of the Gaussian integers requires overcoming additional hurdles which are not present in the naive Sieve of Eratosthenes. Despite this, nearly all of the basic computational sieve ideas can be extended to Gaussian integers. In particular, I extend both the wheel sieve and the segmented sieve.

The [_primesieve_](https://github.com/kimwalisch/primesieve) project implements the current state of the art prime generator for rational integers. _primesieve_ includes both algorithmic optimizations as well as the hardware considerations needed to improve runtime performance. In this repository, I draw inspiration from _primesieve_ and hope to extend some of its algorithms and architectures to the realm of the Gaussian integers.

The motivation behind this project is to develop an API to allow for easy access to the Gaussian integers, an abstract mathematical structure lacking native support in most programming languages. The Gaussian integers provide a rich source of mathematical phenomena which have been well studied in domain of the rational integers. While the essence of mathematics is theorem proving, mathematical examples play an extremely valuable role: they inspire further questions, they guide conjecture, and they inform insight. This project bridges the gap between theory and example in the context of Gaussian primes.

To the best of my knowledge, a highly efficient and accessible sieve has never been implemented in the Gaussian integers. Moreover, tools to access applications, such as Gaussian prime races and the Gaussian moat problem, are not available. This project hopes to provide such resources.

## Install

### Dependencies

Building the source requires the `clang` compiler and the `libc++` library.

On macOS, these dependencies can be obtained by installing the command line tools.

```sh
xcode-select --install
```

On a Linux, these dependencies can be installed through apt.

```sh
sudo apt install clang libc++-dev libc++abi-dev
```

### Build

With these dependencies in place, you can build the C++ executables (this is not needed for Python installation). Clone or download this repository and call the Makefile.

```sh
git clone https://github.com/zebengberg/gaussian-integer-sieve
cd gaussian-integer-sieve
make
```

### Installing the Python API

After installing the [dependencies](#dependencies), the `gaussianprimes` module can be installed through PyPI.

```sh
pip install gaussianprimes
```

This approach has been tested on MacOS X and Ubuntu 20.04.

The [Python API](#python-api) can also be installed from source with `pip`.

```sh
git clone https://github.com/zebengberg/gaussian-integer-sieve
cd gaussian-integer-sieve
make pyinstall
```

After the module is installed, it can be imported into a python console or used in a python script with `import gaussianprimes`.

## Command line usage

All sieving algorithms can be called from the `gintsieve` executable.
Command line options include:

```text
Usage: ./gintsieve x [y dx dy alpha beta] [option1] [option2] ...
Generate Gaussian primes with norm up to x using sieving methods.
    x                   Norm-bound of the generated primes
    y                   Coordinates (x, y) of SW-corner of array in block sieve mode.
    dx                  Horizontal side length in block sieve mode.
    dy                  Vertical side length in block sieve mode.
    alpha               Start angle in sector sieve mode.
    beta                Terminal angle in sector sieve mode.

Options:
    -h, --help          Print this help message.
    -v, --verbose       Display sieving progress.
    -p, --printprimes   Print the real and imag part of primes found by the sieve.
    -w, --write         Write primes to csv file in current directory.
    -a, --printarray    Print a text representation of the sieve array.
    -c, --count         Count the number of generated primes and exit program.

Optional sieve types:
    -o, --octant        Sieve array indexed by Gaussian integers in the first octant.
                        This is the default sieve method called.
    -s, --sector        Sieve array indexed by Gaussian integers in the sector
                        with start angle alpha and final angle beta.
    -b, --block         Sieve array indexed by Gaussian integers in the rectangle
                        defined by x <= real < x + dx and y <= imag < y + dy.
    -d, --donut         If a donut version of the sieve array exists, use it. In the
                        donut sieve, the sieve array consists of Gaussian integers
                        coprime to 2 and 5. This option can be used with --octant
                        and --block, and is often significantly faster.
```

For example, to print the real and imaginary parts of the Gaussian primes up to norm 60 sorted by norm, run:

```sh
$ ./gintsieve 60
1 1
2 1
1 2
3 0
3 2
2 3
4 1
1 4
5 2
2 5
6 1
1 6
5 4
4 5
7 0
7 2
2 7
Total number of primes printed: 17
```

Many of the sieving algorithms can be visualized by calling the `-a` option which prints the _sieve array_ after sieving is complete. For example, below is a hexidecimal encoding of the donut sieve array for primes in the first octant up to norm 12000.

```sh
$ ./gintsieve 12000 -a -d
                                                              d34cdbff ffffffec
                                                     0aedf3ff c0120c78 da730b2e ffffffef
                                            6fdcd7ff 1f08ddf0 eed55590 d18b13c7 ffeebdb2
                                   f65cffef 70113512 369847a3 36968225 e82622d5 1149408b fffffffd
                          43eedbff f8cba42e 51413f89 082e18b5 88face6d 2a0c2722 5316d330 fffff844
                 0bffdfef 0e47b6da 25f0b9da f9245714 87cac0ea 8894d858 8f4c51d5 9602fc1a ff54b816
        f3ceffef ac979b9b be4c61d3 94f6cea5 64332c6d 660ea2b6 1196506d 40fdb31f 2a1e034f f5c0ac56
5fffffee bb73ba7f c9767527 1739d9f0 c3a99523 bf5e23a4 199f8080 f06afb52 554124ac 4f237261 48e00f02
```

To see the primality of an 80 x 20 block of Gaussian integers far out the in complex plane, we run:

```sh
$ ./gintsieve 100000000 300000000 80 20 -a -b
--------------------------------------------*-*-------------------*-------*-----
---------------*---------------------------------------------------*-------*----
--------------------------------*-----------------------------------------------
---------------------*-------------------------------------*-----*---*-*--------
------------------------------------*---------------*---------------------------
-----------------------------*---------------------------------------------*----
--------------------------------*-----------------------------------------------
-------------*-----------*-------------------------------*----------------------
--------------*-----------*-----------------------------*---*-------------------
---*-------*-----------*-------*-----------------*------------------------------
--------------------------------------------------------------------------------
-------------------------*-*----------------------------------------------------
------------------------------*-*-----*-----------------------------------------
---------*-----------------------------------*---*-------------------*----------
------*---------*-----*-------------------------*---------------------------*---
---------------*-----*---*---------------*-------------------------------------*
--------------------------------------------------------------------------------
---*---------*-----------------------------------------*-------*----------------
----------------------------------------------------------------*-----*---*-----
-*-----------------------------*-*---*-*---------------------------*------------
```

## Python API

Once the python module `gaussianprimes` is [built](#install), we can import it into python file or notebook and use it as in the following examples.

```python
>>>  import gaussianprimes as gp

# Generate a numpy array containing coordinates of Gaussian primes up to norm 50 sorted by norm.
>>> gp.gprimes(50)
Gints([[1, 2, 1, 3, 3, 2, 4, 1, 5, 2, 6, 1, 5, 4, 7],
       [1, 1, 2, 0, 2, 3, 1, 4, 2, 5, 1, 6, 4, 5, 0]], dtype=int32)

# Instead of returning an array, we can count the Gaussian primes.
>>> gp.count(3141592653)
Calling the OctantSieve to generate smallPrimes...
Building donut sieve array...
Sieve array approximate memory use: 49MB.
Sieve array total number of entries: 394936384
Starting to sieve...
....................................................
Done sieving. Total time for sieving: 12.538 seconds.
Counting primes after sieve...
Total number of primes, including associates: 603726004

603726004

# Plotting Gaussian primes in a rectangular block.
>>> p = gp.gprimes_block(123456, 67890, 100, 100)
>>> p.plot()
```

![Segmented Block](assets/block.png)

```python
# Plot the Gaussian primes in the first octant with norm up to 10000.
>>> p = gp.gprimes_sector(10000, 0, 3.14159 / 8)
>>> p.plot()
```

![First Octant](assets/first_octant.png)

```python
# Plot the Gaussian primes as well as their associates with norm up to 1000.
>>> p = gp.gprimes(1000)
>>> p.plot(full_disk=True)
```

![Full Plane](assets/full_plane.png)

See [this Jupyter notebook](python/examples.ipynb) for many more usage examples.

## Algorithm

Prime generating sieves often employ a _sieve array_ A and a set of _small primes_ P. For each prime p in P, sieving proceeds by crossing off multiples of p in the sieve array A. On a computer, a sieve array can be realized as container of booleans indexed by elements of A. The state of each boolean records if the underlying index has been crossed off by some prime p in P.

In broad strokes, sieving in the Gaussian integers involves the following steps.

1. Initialize a 2-dimensional array A with index (a, b) corresponding to the Gaussian integer a + bi. In this way, we can view this array as representing a finite subset of the Gaussian integers. Multiplication by i gives a 4-fold rotational symmetry in the complex plane and complex conjugation (in conjunction with multiplication by i) gives a reflectional symmetry about the line y = x in the complex plane. Such symmetries allow us to reduce the size of the sieving array if it contains redundancies encapsulated by these symmetries.

2. Generate a set of _small primes_ P either through calling another sieve or by using the existing sieve array to locate uncrossed elements of small norm. The set P contains all primes with norm up to sqrt(B), where B is the norm of the largest Gaussian integer in A. In either case, for each prime a + bi in P, leave the sieve array A uncrossed at a + bi but mark off every Gaussian integer multiple of a + bi. Continue this process for each prime in P.

3. The unmarked entries that remain in the array A are exactly the Gaussian primes in A.

[See here](assets/motivation.pdf) for a discussion of different approaches to generating primes in the Gaussian integers and some motivation for our choice of algorithm.

### Crossing off multiples

Given a Gaussian integer a + bi, we are tasked with crossing off multiples of a + bi from our sieve array. A multiple of a + bi is of the form (a + bi)(c + di) for c, d in Z. Expanding this, we seek indices of the form (u, v) where
u = ac - bd, and v = ad + bc.
This simply requires a double for-loop (once over c and once over d). Moreover, we can avoid costly multiplications by adding or subtracting c's or d's from the current index u or v while iterating. In the language of abstract algebra, we are crossing off elements of the Z-module generated by (a, b) and (-b, a).

### Donut Sieve

In the rational integers Z, the [wheel sieve](https://en.wikipedia.org/wiki/Wheel_factorization) allows for a more efficient way to manage and step through the sieve array. Primes other than 2, 3, and 5 must lie in one of the 8 distinct residue classes 1, 7, 11, 13, 17, 19, 23, 29 mod 30. In light of this observation, the wheel sieve (with a wheel size of 30) reduces both the memory and complexity of sieving by only tracking these 8 distinguished residue classes. If implemented carefully, a wheel sieve will not add significant overhead to the actual sieving process, and it will reduce to the total size of the sieve array to 8 / 30 = 27% of the original size.

This project extends the idea of a wheel sieve to the Gaussian integers. Instead of a wheel corresponding to the quotient Z/30Z rolling over the integers, we consider the finite set of remainders in Z[i]/10Z[i]. Mathematically, this quotient is a **donut** or torus. Just as the primes 2, 3, 5 in the rational integers give rise to a mod-30 wheel, the mod-10 donut arises from the primes 1 + i, 1 + 2i and 2 + i. The first of these three primes sits above the ramifying prime 2, and the second and third both sit above the splitting prime 5.

- Every Gaussian integer z satisfying z = 0 (mod 1 + i) should be omitted from the sieve array. The prime 1 + i has norm 2, and so one out of every two Gaussian integers will be divisible by 1 + i. In this way, including 1 + i in the donut-base will reduce the sieve array size by 50%.
- We deal with the primes 1 + 2i and 2 + i together (by the Chinese Remainder Theorem, this is equivalent to working in Z[i]/5Z[i]). Every Gaussian integer satisfying z = 0 (mod 1 + 2i) **or** z = 0 (mod 2 + i) should be omitted from the sieve array. By the inclusion-exclusion principle and the CRT, there are exactly 9 solutions to these equations in Z[i]/5Z[i]. The Gaussian integer 5 has norm 25, and so 9 out of every 25 Gaussian integers will be removed from the sieve array. This provides a savings of 9/25 = 36%.
- Combining both of these local conditions, our donut allows us to only consider 32 of the 100 residue classes in Z[i]/10Z[i].

In the wheel sieve, adding more primes to the wheel-base will allow more savings within the sieve array at the expense of increasing the combinatorial complexity of the sieving process itself. We stopped with a donut of size 10 for hardware-related reasons (see [implementation](#c-implementation)), however, the following additional primes could be consider.

- The inert prime 3 in the donut would give a savings of 1/9 = 11% in sieve array size.
- The primes 3 + 2i and 3 + 2i, those dividing 13, would give a savings of 25/169 = 15% in sieve array size.

In the classes `OctantDonutSieve` and `BlockDonutSieve` we implement the basic structure of the donut sieve. In particular, these classes contain variables and methods for accessing and dealing with the donut. The array below exemplifies how to deal with donut difficulties in a mod-10 donut. The entries in this array correspond to horizontal gaps between distinguished residue classes.

```text
{{0, 2, 0, 4, 0, 0, 0, 2, 0, 2},
 {4, 0, 0, 0, 2, 0, 4, 0, 0, 0},
 {0, 0, 0, 2, 0, 2, 0, 6, 0, 0},
 {2, 0, 6, 0, 0, 0, 0, 0, 2, 0},
 {0, 4, 0, 0, 0, 4, 0, 0, 0, 2},
 {0, 0, 2, 0, 2, 0, 2, 0, 4, 0},
 {0, 4, 0, 0, 0, 4, 0, 0, 0, 2},
 {2, 0, 6, 0, 0, 0, 0, 0, 2, 0},
 {0, 0, 0, 2, 0, 2, 0, 6, 0, 0},
 {4, 0, 0, 0, 2, 0, 4, 0, 0, 0}}
```

### Segmented Sieve

In a segmented sieve, we break up the sieving array into smaller subsets called **segments**. With these segments in place, we perform sieving (crossing off multiples of small primes) segment by segment. There are advantages and disadvantages to this segmented approach.

- The main advantage of segmentation is that each segment can fit more readily into computer hardware such as the L2-cache. This will allow the inner for-loop in the snippet below to execute more quickly because each segment can be quickly accessed.
- The main disadvantage of segmentation is some additional overhead. Using code snippet below as an example, for some primes `p` there will be few or no calls to the function `cross_off_multiples`. As a rule of thumb, each segment should contain at least one multiple of each small prime `p`.

```python
for segment in sieve_array:
    for p in small_primes:
        cross_off_multiples(segment, p)
```

In this project, segmentation can be achieved by calling instances of the `BlockSieve` class. In `VerticalMoat` and `SegmentedMoat`, we take this approach to explore Gaussian primes.

## C++ Implementation

The aforementioned algorithm is implemented in a C++ library. `BaseSieve` is an abstract base class with some basic sieving methods. Classes derived from this include `OctantSieve`, `OctantDonutSieve`, `SectorSieve`, `BlockSieve`, and `BlockDonutSieve`. Each derived class has its own method for initiating and accessing the sieve array. See the [usage examples](#command-line-usage) for various text representations of these sieve arrays.

In classes `OctantSieve`, `SectorSieve`, and `BlockSieve`, the 2-dimensional sieve array A is stored with a `vector<vector<bool>>` container. This C++ object is memory-efficient: each boolean is stored as a single bit in memory (this in itself gives a 224-fold savings over python 3.7 which requires 28 bytes of memory to store a boolean value).

In implementing the donut sieve in the classes `OctantDonutSieve` and `BlockDonutSieve`, each 10 x 10 block of Gaussian integers corresponds to a full _donut roll_. The [donut sieve](#donut-sieve) requires holding 32 residue classes for every 10 x 10 block of Gaussian integers. Said differently, every 10 x 10 block of Gaussian integers requires 32 bits of information to store its current state in the sieve process. Conveniently, a C++ `int` typically also requires 32 bits of memory space. In this way, in donut-based classes our sieve array is a `vector<vector<int>>` container in C++.

## Applications

This library can be used to generate data that sheds new light on several unsolved problems in number theory.

- [Gaussian prime races in sectors](#gaussian-prime-races-in-sectors)
- [Angular distribution of Gaussian primes](#angular-distribution-of-gaussian-primes)
- [The Gaussian moat problem](#the-gaussian-moat-problem)

All three of these applications have implementations within the [Python API](#python-api) and can be found inside the `gintsieve` python module. [This Jupyter notebook](python/examples.ipynb) gives some examples.

### Gaussian prime races in sectors

Prime number races have been thoroughly studied beginning with an observation of Chebyshev in 1853. Odd prime numbers fall into two disjoint categories: primes of the form 4k + 1, and primes of the form 4k + 3. Chebyshev noted that when counting the number of primes in these two sets up to some threshold x, the primes of the form 4k + 3 often seemed to lead the race. This phenomenon has been extensively quantified and generalized in the realm of the rational integers, and is known as _Chebyshev bias_.

In the Gaussian integers, rather than group a prime by its remainder mod 4, we can categorize Gaussian primes by sector. A [classical result](http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002365162) of Hecke states that the angles determined by Gaussian primes are _equidistributed_. In other words, given two sectors in the complex plane with equal central angles, Hecke proved that as the radius of the sector grows large, both have the same asymptotic number of primes. Just as Chebyshev raced the primes in the residue classes 4k + 1 and 4k + 3, one can hold a race among Gaussian primes in two sectors with equal central angle.

The `SectorSieve` class performs sieving in a specified sector in the complex plane. We use this class to conduct races within sectors and calculate Chebyshev bias. To the best of my knowledge, these prime number races in sectors were previously unstudied. The [Python API](#python-api) contain a class for exploring these sector races.

### Angular distribution of Gaussian primes

In addition to two-way prime number races, one can consider races with many participants. In the rational primes, odd primes fall into one of the four categories 8k + 1, 8k + 3, 8k + 5, and 8k + 7. As a consequence, we can consider the four-way race between these four disjoint residue classes mod 8. This many-way race can just as easily be considered with Gaussian primes. In particular, for a collection of sectors having the same central angle, we consider the distribution of the counts of Gaussian primes within these sectors as the bound on the sectors' radii grows. This is data collection is implemented in the [Python API](#python-api).

### The Gaussian moat problem

To conceptualize the Gaussian moat problem, imagine a frog jumping between lily pads situated in the complex plane at exactly the Gaussian primes. The frog, having finite strength, can only take jumps of bounded size. A famous unsolved problems asks if such a frog could travel arbitrarily far from its starting point. If this frog fails to do so, it must have encountered some _Gaussian moat_ preventing it from making progress.

This problem, lying at the interface between number theory and percolation theory, has been extensively studied since it was first posed in 1962. In 2004, Tsuchimura computed the state of the art result by showing that a frog starting at the origin and taking jumps of distance at most 6 will encounter a Gaussian moat. It is a folklore conjecture that given any bounded jump size, a moat will always be encountered.

Beginning with a bound on the frog's jump, form a graph whose vertices are Gaussian primes. Two vertices are adjacent if the distance between them is less than or equal to the jump bound. Three classes within this library explore the _main connected component_ of this graph, that is, the connected component containing the Gaussian prime 1 + i (the prime closest to the origin).

- In the class `OctantMoat`, the main connected component is explored using a _depth-first search_ approach. Initially, all primes are generated and held in memory using the `OctantSieve` class. Once the sieve array is larger than the memory capabilities of the computer calling this class, this approach fails.
- In `SegmentedMoat`, the same component is explored using a segmented sieving approach. Instances of `BlockSieve` are called and explored; boundary data is passed from one instance to the next. This approach is to similar to the algorithm in Tsuchimura's paper _Computational Results for Gaussian Moat Problem_.
- In `VerticalMoat`, we consider a vertical strip in the complex plane running between the real-axis and the diagonal line y = x. As in `SegmentedMoat`, instances of `BlockSieve` are called and boundary data is shared. This algorithm searches for a sequence of adjacent Gaussian primes spanning the vertical strip. If no such sequence is found, then a moat is present, and as a consequence, the main component is finite. A similar approach was taken in the paper _A Stroll Through the Gaussian Primes_ by Gethner, Wagon, and Wick.

All of these moat-exploring classes can be accessed from the `gintmoat` executable. Command line options include:

```text
Usage: ./gintmoat jumpSize [realPart] [option1] [option2] ...
Explore the connected component at the origin in the Gaussian moat problem.
    jumpSize            The jump bound giving adjacency relation among primes.
    realPart            The real-part of the vertical strip to explore if in
                        vertical mode.

Exploration modes:
    --origin            Explore the connected component starting at the origin until
                        an impassable moat is encountered. Default exploration mode.
    --segmented         Use a segmented approach to explore the connected component.
                        Algorithm is similar to that in Tsuchimura paper. This approach
                        only counts the size of the component.
    --vertical          Search for a Gaussian moat along a thin vertical strip starting
                        at real-part x. Used to show a component is finite.

Options:
    -h, --help          Print this help message.
    -v, --verbose       Display progress.
    -p, --printprimes   Print the real and imag part of primes in the connected component
                        if in origin mode.
```

For example, to print all primes that can be reach with jumps up to distance 1.5, we run:

```sh
$ ./gintmoat 1.5 -p

1 1
2 1
3 0
3 2
4 1
5 2
6 1
7 0
7 2
8 3
9 4
8 5
10 3
11 4
The main connected component has size: 14
The farthest out prime in component has coordinates: 11 4
```

To get the size of the component with jumps up to distance 4.5, we could run the following (and wait nearly an hour):

```sh
$ ./gintmoat 4.5 --segmented


The main connected component has size: 273791623
```

Many other graph theoretic statistics could be gathered using a similar implementation. Examples of interesting data to consider include:

- The distribution of the vertex degrees in the main component
- Number and location of _bridges_
- Size of other connected components (away from the origin)
- "Lakes" and other regions void of primes

This project could readily be adapted to explore such phenomena.

## Tests

### Python

Test the `gaussianprimes` Python package by running `python python/test_gaussianprimes.py`.

### C++

Results of this repository can be verified with [http://oeis.org/A091100](http://oeis.org/A091100). The program `ginttest.cpp` automates this.

```sh
make test
```

Markdown-formatted output of this is pasted below.

#### Testing and timing OctantSieve and OctantDonutSieve

| norm bound | # of primes including associates | OctantSieve time | OctantDonutSieve time |
| ---------- | -------------------------------- | ---------------- | --------------------- |
| 2^20       | 327972                           | 0.039 s          | 0.017 s               |
| 2^21       | 622700                           | 0.073 s          | 0.032 s               |
| 2^22       | 1184128                          | 0.147 s          | 0.065 s               |
| 2^23       | 2256140                          | 0.295 s          | 0.13 s                |
| 2^24       | 4311256                          | 0.622 s          | 0.283 s               |
| 2^25       | 8254712                          | 2.247 s          | 0.551 s               |
| 2^26       | 15830020                         | 2.8 s            | 0.953 s               |
| 2^27       | 30415816                         | 8.317 s          | 3.281 s               |
| 2^28       | 58525604                         | 12.048 s         | 3.927 s               |
| 2^29       | 112770284                        | 26.179 s         | 8.516 s               |
| 2^30       | 217599824                        | 56.611 s         | 18.362 s              |

#### Testing and timing BlockSieve and BlockDonutSieve with random rectangles

| block                                       | # of primes | BlockSieve time | BlockDonutSieve time |
| ------------------------------------------- | ----------- | --------------- | -------------------- |
| [50000000, 50000700) x [80000000, 80000600) | 14599       | 8.787 s         | 4.456 s              |
| [80000000, 80000300) x [20000000, 20000100) | 1069        | 6.239 s         | 4.261 s              |
| [30000000, 30000600) x [40000000, 40000100) | 2152        | 5.006 s         | 2.783 s              |
| [50000000, 50000100) x [80000000, 80000900) | 3137        | 9.424 s         | 5.808 s              |
| [90000000, 90000200) x [90000000, 90000800) | 5432        | 11.09 s         | 5.532 s              |
| [20000000, 20000100) x [90000000, 90000100) | 352         | 6.33 s          | 3.875 s              |
| [30000000, 30000300) x [50000000, 50000500) | 5370        | 3.868 s         | 2.502 s              |
| [40000000, 40000500) x [20000000, 20000700) | 12661       | 3.215 s         | 2.008 s              |
| [10000000, 10000200) x [80000000, 80000600) | 4209        | 5.864 s         | 3.595 s              |
| [30000000, 30000200) x [20000000, 20000400) | 2915        | 2.609 s         | 1.714 s              |

#### Testing and timing SectorSieve with random sectors

| alpha     | beta     | beta - alpha | norm bound | # of primes | time   |
| --------- | -------- | ------------ | ---------- | ----------- | ------ |
| 0.228457  | 0.75373  | 0.525273     | 2^20       | 27410       | 0.041  |
| 0.54119   | 0.763206 | 0.222016     | 2^21       | 22008       | 0.038  |
| 0.768652  | 0.773941 | 0.00528918   | 2^22       | 971         | 0.014  |
| 0.508     | 0.665647 | 0.157647     | 2^23       | 56579       | 0.101  |
| 0.0176164 | 0.636602 | 0.618986     | 2^24       | 424631      | 0.691  |
| 0.257315  | 0.58076  | 0.323445     | 2^25       | 424979      | 0.759  |
| 0.600125  | 0.764335 | 0.16421      | 2^26       | 413633      | 0.793  |
| 0.099027  | 0.560295 | 0.461268     | 2^27       | 2233083     | 5.242  |
| 0.254282  | 0.622513 | 0.368231     | 2^28       | 3429942     | 8.611  |
| 0.671402  | 0.766183 | 0.0947814    | 2^29       | 1700925     | 4.688  |
| 0.0379932 | 0.731988 | 0.693995     | 2^30       | 24034855    | 89.962 |

#### Timing SectorSieve with random thin sectors

| alpha    | beta     | beta - alpha | norm bound | # of primes | time   |
| -------- | -------- | ------------ | ---------- | ----------- | ------ |
| 0.542557 | 0.543533 | 2^-10        | 2^30       | 34014       | 0.548  |
| 0.165816 | 0.166304 | 2^-11        | 2^31       | 32729       | 0.866  |
| 0.321085 | 0.32133  | 2^-12        | 2^32       | 31491       | 1.319  |
| 0.419426 | 0.419548 | 2^-13        | 2^33       | 30626       | 2.227  |
| 0.341336 | 0.341397 | 2^-14        | 2^34       | 29533       | 3.361  |
| 0.65282  | 0.652851 | 2^-15        | 2^35       | 28798       | 5.728  |
| 0.258526 | 0.258541 | 2^-16        | 2^36       | 27799       | 9.022  |
| 0.415701 | 0.415708 | 2^-17        | 2^37       | 27117       | 13.897 |
| 0.300503 | 0.300507 | 2^-18        | 2^38       | 26630       | 23.102 |
| 0.247465 | 0.247466 | 2^-19        | 2^39       | 25726       | 37.014 |
| 0.422951 | 0.422952 | 2^-20        | 2^40       | 24981       | 54.84  |

#### Testing OctantMoat and SegmentedMoat

| jumpSize | size of main component | farthest prime encountered |
| -------- | ---------------------- | -------------------------- |
| 2        | 92                     | 42 17                      |
| 3        | 380                    | 84 41                      |
| 3.5      | 31221                  | 976 311                    |
| 4        | 347638                 | 3297 2780                  |
| 4.3      | 2386129                | not computed               |

## License

[MIT License](LICENSE.md)