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https://github.com/antononcube/raku-math-numbertheory

Raku package with number theory functions.
https://github.com/antononcube/raku-math-numbertheory

euler-phi modular-exponentiation modular-inverse number-theory prime-factors raku rakulang totient

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Raku package with number theory functions.

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# Math::NumberTheory



Raku package with Number theory functions.



The function names and features follow the 

[Number theory functions of Wolfram Language](http://reference.wolfram.com/language/guide/NumberTheory.html).



**Remark:**  Raku has some nice built-in Number theory functions, like, `base`, `mod`, `polymod`, `expmod`, `is-prime`. 

They somewhat lack generality, hence their functionalities are extended with this package. 

For example, `is-prime` works with lists and [Gaussian integers](https://en.wikipedia.org/wiki/Gaussian_integer).



-------



## Installation



From [Zef ecosystem](https://raku.land):



```

zef install Math::NumberTheory

```



From GitHub:



```

zef install https://github.com/antononcube/Raku-Math-NumberTheory

```



-------



## Usage examples



### Prime number testing



The built-in sub `is-prime` is extended to work with [Gaussian integers](https://en.wikipedia.org/wiki/Gaussian_integer):



```perl6

use Math::NumberTheory;

say is-prime(2 + 1i);

say is-prime(5, :gaussian-integers);

```

```

# True

# False

```



Another extension is threading over lists of numbers:



```perl6

is-prime(^6)

```

```

# (False False True True False True)

```



### Factor integers



Gives a list of the prime factors of an integer argument, together with their exponents:



```perl6

factor-integer(factorial(20))

```

```

# [(2 18) (3 8) (5 4) (7 2) (11 1) (13 1) (17 1) (19 1)]

```



**Remark:** By default `factor-integer` uses [Pollard's Rho algorithm](https://en.wikipedia.org/wiki/Pollard's_rho_algorithm) --

specified with `method => 'rho'` --

as implemented at RosettaCode, [RC1], with similar implementations provided by "Prime::Factor", [SSp1], and "Math::Sequences", [RCp1].



Do partial factorization, pulling out at most `k` distinct factors:



```perl6

factor-integer(factorial(20), 3, method => 'trial')

```

```

# [(2 18) (3 8) (5 4)]

```



### Chinese reminders



Data:



```perl6

my @data = 931074546, 117172357, 482333642, 199386034, 394354985;

```

```

# [931074546 117172357 482333642 199386034 394354985]

```



Keys:



```perl6

#my @keys = random-prime(10**9 .. 10**12, @data.elems);

my @keys = 274199185649, 786765306443, 970592805341, 293623796783, 238475031661;

```

```

# [274199185649 786765306443 970592805341 293623796783 238475031661]

```



**Remark:** Using these larger keys is also a performance check.



Encrypted data:



```perl6

my $encrypted = chinese-reminder(@data, @keys);

```

```

# 6681669841357504673192908619871066558177944924838942629020

```



Decrypted:



```perl6

my @decrypted = @keys.map($encrypted mod *);

```

```

# [931074546 117172357 482333642 199386034 394354985]

```



### Modular exponentiation and modular inversion



The sub `power-mod` extends the built-in sub `expmod`.

The sub `modular-inverse` is based on `power-mod`.



`expmod` gives an error and no result when the 1st argument cannot be inverted with the last argument:



```perl6

expmod(30, -1, 12)

```

```

#ERROR: Error in mp_exptmod: Value out of range

# Nil

```



`power-mod` returns `Nil`:



```perl6

power-mod(30, -1, 12).defined

```

```

# False

```



### Number-base related



There are several subs that provide functionalities related to number systems representation.



For example, here we find the digit-breakdown of $100!$:



```perl6

100.&factorial.&digit-count

```

```

# {0 => 30, 1 => 15, 2 => 19, 3 => 10, 4 => 10, 5 => 14, 6 => 19, 7 => 7, 8 => 14, 9 => 20}

```



Here is an example of using `real-digits`:



```perl6

real-digits(123.55555)

```

```

# ([1 2 3 5 5 5 5 5] 3)

```



Non-integer bases can be also used:



```perl6

my $r = real-digits(π, ϕ);

```

```

# ([1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1] 3)

```



Here we recover $\pi$ from the Golden ratio representation obtained above:



```perl6

$r.head.kv.map( -> $i, $d { $d * ϕ ** ($r.tail - $i - 1)  }).sum.round(10e-12);

```

```

# 3.1415926535899996

```



The sub `phi-number-system` can be used to compute

[Phi number system](https://mathworld.wolfram.com/PhiNumberSystem.html)

representations:



```perl6

.say for (^7)».&phi-number-system

```

```

# ()

# (0)

# (1 -2)

# (2 -2)

# (2 0 -2)

# (3 -1 -4)

# (3 1 -4)

```



**Remark:** Because of the approximation of the Golden ratio used in “Math::NumberTheory”,

in order to get exact integers from phi-digits we have to round using small multiples of 10. 



-------



## References



[RC1] Rosetta Code, [Prime decomposition](https://rosettacode.org/wiki/Prime_decomposition),

[Section "Pure Raku"](https://rosettacode.org/wiki/Prime_decomposition#Pure_Raku).



[RCp1] Raku Community,

[Math::Sequences Raku package](https://github.com/raku-community-modules/Math-Sequences),

(2016-2024),

[GitHub/raku-community-modules](https://github.com/raku-community-modules).



[SSp1] Stephen Schulze,

[Prime::Factor Raku package](https://github.com/thundergnat/Prime-Factor),

(2016-2023),

[GitHub/thundergnat](https://github.com/thundergnat).