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توضیحات

From A to Z Number Theory

ویژگی	مقدار
سیستم عامل	-
نام فایل	aznt-0.0.41
نام	aznt
نسخه کتابخانه	0.0.41
نگهدارنده	[]
ایمیل نگهدارنده	[]
نویسنده	-
ایمیل نویسنده	"Adrian Zapała, MSc" <adrian.zapala@outlook.com>
آدرس صفحه اصلی	-
آدرس اینترنتی	https://pypi.org/project/aznt/
مجوز	-

![Python Version](https://img.shields.io/badge/python-3.11-blue/) ![aznt version](https://img.shields.io/badge/aznt-0.0.41-green) ![Beta version](https://img.shields.io/badge/beta-ver.-green) # `aznt` - From A to Z Number Theory * Number Theory library for Python with prime numbers. * **Important:** requires Python >= 3.11. * You can pass to author's GitHub at [adrianzapala.github.io](https://adrianzapala.github.io/). # Install ```python pip install aznt ``` # Imports The package `aznt` includes three modules: `azntnumbers`, `azntprimality` and `azntplots`. **Examples** ```python import aznt.azntnumbers as azn import aznt.azntprimality as azpr import aznt.azntplots as azpt ``` # General info about performance When working with numbers with a large number of digits, calculation results slow down quickly. This is especially true for algorithms such as factorization, searching for dividers etc. So for many large numbers, it can take a very long time to get results. # Usage: `azntnumbers` module functions ## `factorization(n)` Return prime factors of a number from a function argument. Time complexity: `O(n)`. **Examples** ```python >>> factorization(8) [2, 2, 2] >>> factorization(1123243) [11, 11, 9283] >>> factorization(20570952) [2, 2, 2, 3, 17, 127, 397] ``` ## `dividers_naive(n)` Returns divisors of a number from a function argument. It returns a list of these divisors. Time complexity: `O(n)`. **Examples** ```python >>> dividers_naive(1123243) [1, 11, 121, 9283, 102113, 1123243] >>> dividers_naive(20570952) [1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 127, 136, 204, 254, 381, 397, 408, 508, 762, 794, 1016, 1191, 1524, 1588, 2159, 2382, 3048, 3176, 4318, 4764, 6477, 6749, 8636, 9528, 12954, 13498, 17272, 20247, 25908, 26996, 40494, 50419, 51816, 53992, 80988, 100838, 151257, 161976, 201676, 302514, 403352, 605028, 857123, 1210056, 1714246, 2571369, 3428492, 5142738, 6856984, 10285476, 20570952] ``` ## `dividers_opt(n)` This function calculates divisors in pairs, which multiplication product creates a number from function argument. If the argument is default (`pairs=False`) and also if the numbers from these pairs are different it puts these pairs on the list and then sort list (if the numbers are equal, it's placing only one number on the list). Alternatively, if the argument is `True` it join pairs of divisors (may be equal) into tuples and then append them into a list. This function works pretty much faster than `dividers_naive()` function. Time complexity: `O(sqrt(n)`. **Examples** ```python dividers_opt(100) [1, 2, 4, 5, 10, 20, 25, 50, 100] dividers_opt(100, pairs=True) [(1, 100), (2, 50), (4, 25), (5, 20), (10, 10)] ``` ## `tau(n)` Count the number of divisors of an integer (including 1 and the number itself). Time complexity: `O(sqrt(n))`. **Examples** ```python >>> tau(6) 4 >>> tau(4356) 27 >>> tau(12499674) 16 ``` ## `sigma(n)` Return the sum of divisors of an integer (including 1 and the number itself). Time complexity: `O(sqrt(n)))`. **Examples** ```python >>> sigma(3) 4 >>> sigma(343532) 687120 ``` ## `s(n)` Return the sum of proper divisors of an integer (excluding `n` itself). Time complexity: `O(sqrt(n)))`. **Examples** ```python >>> s(3) 1 >>> s(4438) 3194 ``` ## `gcd_mod(a, b)` Return Greatest Common Divisor (GCD) of two positive integers using Euclidean algorithm with division. Time complexity: `O(log n))`. **Examples** ```python >>> gcd_mod(36, 882) 18 >>> gcd_mod(363, 1287) 33 >>> gcd_mod(23, 3456) 1 >>> gcd_mod(484, 56) 4 ``` ## `gcd_subtract(a, b)` Return Greatest Common Divisor (GCD) of two positive integers using Euclidean algorithm with subtraction. Time complexity: `O(n))`. **Examples** ```python >>> gcd_subtract(36, 882) 18 >>> gcd_subtract(363, 1287) 33 >>> gcd_subtract(23, 3456) 1 >>> gcd_subtract(484, 56) 4 ``` ## `lcm(a, b)` Return Lowest Common Multiple (LCM) of two positive integers. Time complexity: `O(log n))`. **Examples** ```python >>> lcm(36, 882) 1764 >>> lcm(363, 1287) 14157 >>> lcm(23, 3456) 79488 >>> lcm(484, 56) 6776 ``` ## `is_gcd_eq_1(n, floor, ceil)` Return a tuple. First element of a tuple is a quantity of pairs of numbers which are relatively prime to themselves, the second element is a density of them and the third element is a list of nested lists of them. All pairs are randomly generated from `floor` to `ceil` of numbers range (`n`). Based on that a variety of different outputs can be possible for the same parameters. Time complexity: `O(n log n))`. **Examples** ```python >>> is_gcd_eq_1(25, 100, 10000) (15, 0.6, [[1470, 8779], [8447, 5117], [7852, 5597], [8111, 7225], [5375, 6721], [9191, 4239], [7101, 6361], [1099, 211], [7036, 9515], [5281, 7212], [948, 2471], [3451, 4541], [3896, 9463], [8006, 845], [4448, 4803]]) >>> is_gcd_eq_1(25, 100, 10000) (13, 0.52, [[7271, 5402], [9505, 5791], [8608, 1811], [6017, 9782], [1784, 3439], [8135, 9444], [3979, 9078], [2849, 2677], [888, 8171], [6399, 6145], [6647, 3679], [3697, 7051], [557, 553]]) ``` ## `totient(n)` Return Euler's totient function (phi) which counts the positive integers up to a given integer `n` that are relatively prime to `n`. Time complexity: `O(log n))`. **Examples** ```python >>> totient(4) 2 >>> totient(24) 8 >>> totient(424235) 253440 ``` ## `pnt(n)` Return the asymptotic distribution of the prime numbers among the positive integers. Time complexity: `O(log n)`. **Examples** ```python >>> pnt(1000) 144.76482730108395 // Real value: 168 >>> pnt(1000000) 72382.41365054197 // Real value: 78498 ``` ## `prime_probability(n)` Return probability if a random number is prime to a specific ceil given. Time complexity: `O(log n)`. **Examples** ```python >>> prime_probability(100) 0.21714724095162588 >>> prime_probability(1000) 0.14476482730108395 ``` ## `basel_problem(max, n=2)` Return precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series. The sum of the series is approximately equal to 1.644934 (pi^2 / 6 for `n` = 2). Time complexity: `O(n)`. **Examples** ```python >>> basel_problem(345567) 1.6449311730576142 >>> basel_problem(2568, n=4) 1.0823232336914694 ``` ## `is_mersenne(m)` Check if a number is Mersenne number, M = 2^n - 1 (prime or composite). Returns a four-element tuple. First element is boolean value - `True` or `False`. If `True`, then second argument is M with exponent, third is the number in exponent (scientific) notation and the last is whole number. Time complexity: `O(log n)`. **Examples** ```python >>> is_mersenne(0) Argument: 0. Mersenne numbers starts from 1. >>> is_mersenne(1) (True, 'M1', '1.00e+00', 1) >>> is_mersenne(2) False >>> is_mersenne(7) (True, 'M3', '7.00e+00', 7) >>> is_mersenne(100) False >>> is_mersenne(255) (True, 'M8', '2.55e+02', 255) >>> is_mersenne(511) (True, 'M9', '5.11e+02', 511) >>> is_mersenne(1023) (True, 'M10', '1.02e+03', 1023) >>> is_mersenne(604462909807314587353087) (True, 'M79', '6.04e+23', 604462909807314587353087) ``` ## `is_fermat(f)` Check if a number is Fermat number, F = 2^2^n - 1 (prime or composite). Returns a four-element tuple. First element is boolean value - `True` or `False`. If `True`, then second argument is F with exponent, third is the number in exponent (scientific) notation and the last is whole number. Time complexity: `O(log (log n))`. **Examples** ```python >>> is_fermat(-3) Argument: -3. Fermat numbers starts from 3. >>> is_fermat(1) Argument: 1. Fermat numbers starts from 3. >>> is_fermat(3) (True, 'F0', '3.00e+00', 3) >>> is_fermat(5) (True, 'F1', '5.00e+00', 5) >>> is_fermat(7) False >>> is_fermat(17) (True, 'F2', '1.70e+01', 17) >>> is_fermat(257) (True, 'F3', '2.57e+02', 257) >>> is_fermat(65537) (True, 'F4', '6.55e+04', 65537) >>> is_fermat(4294967297) (True, 'F5', '4.29e+09', 4294967297) >>> is_fermat(18446744073709551617) (True, 'F6', '1.84e+19', 18446744073709551617) >>> is_fermat(340282366920938463463374607431768211457) (True, 'F7', '3.40e+38', 340282366920938463463374607431768211457) >>> is_fermat(115792089237316195423570985008687907853269984665640564039457584007913129639937) (True, 'F8', '1.16e+77', 115792089237316195423570985008687907853269984665640564039457584007913129639937) >>> is_fermat(13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097) (True, 'F9', '1.34e+154', 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097) ``` ## `is_perfect(n)` Check if a number is perfect number. If so, returns True, False otherwise. Time complexity: `O(sqrt(n)`. **Examples** ```python >>> is_perfect(7) False >>> is_perfect(8128) True ``` ## `zeta_trivial_value(s, n=10000)` Return **trivial** (real) value of Riemann zeta function for given argument, where `s` is zeta function argument, and `n` is number of terms in the series of zeta. Time complexity: `O(n)`. **Examples** ```python >>> zeta_trivial_value(-49) -1.5001733492153957e+23 >>> zeta_trivial_value(-31) 472384867.72163075 >>> zeta_trivial_value(-19) 26.45621212121214 >>> zeta_trivial_value(-15) 0.44325980392156883 >>> zeta_trivial_value(-5) -0.003968253968253967 >>> zeta_trivial_value(-4) 0 >>> zeta_trivial_value(0) 0.5 >>> zeta_trivial_value(.1) -0.6029256240132325 >>> zeta_trivial_value(.5) -1.4482837427744006 >>> zeta_trivial_value(1) inf >>> zeta_trivial_value(5) 1.0369277551433702 >>> zeta_trivial_value(33) 1.0000000001164155 >>> zeta_trivial_value(50) 1.0000000000000009 ``` ## `nontrivial_zeros_dist(t)` Return average distance between zeros on a critical line in Riemann zeta function. Time complexity: `O(log n)`. **Examples** ```python >>> nontrivial_zeros_dist(20) 5.426572570049043 >>> nontrivial_zeros_dist(100) 2.2705167236263177 >>> nontrivial_zeros_dist(1000) 1.2393168126993492 ``` ## `pstats1(data_rows=29)` Return statistics of prime numbers up to `data_rows` which in default is equal `29`. This value can be changed from a function argument, but it can't be greater than that value. It can't be lower than 1 also. Rows statistics starts from 10 up to 10^29, and they are for prime numbers in these ranges. Time complexity: `O(n)`. **Examples** ```python >>> pstats1() <Returns all table> >>> pstats1(-2) Incorrect argument >>> pstats1(5) x π(x) x/π(x) logx (x/π(x) - logx) / x/π(x) [%] 1e+01 4 2.5 2.302585092994046 7.896596280238163 1e+02 25 4.0 4.605170185988092 15.129254649702295 1e+03 168 5.9523809523809526 6.907755278982137 16.050288686899894 1e+04 1229 8.136696501220504 9.210340371976184 13.195083171587305 1e+05 9592 10.42535446205171 11.512925464970229 10.431981059994436 ``` ## `pstats2(data_rows=29)` Return statistics of prime numbers up to `data_rows` which in default is equal `29`. This value can be changed from a function argument, but it can't be greater than that value. It can't be lower than 1 also. Rows statistics starts from 10 up to 10^29, and they are for prime numbers in these ranges. Time complexity: `O(n)`. **Examples** ```python >>> pstats2() <Returns all table> >>> pstats2(-2) Incorrect argument >>> pstats2(4) x PNT: Li(x) π(x) PNT: x/logx 1e+01 6.165599504787298 4 4.3429448190325175 1e+02 30.12614158407963 25 21.71472409516259 1e+03 177.60965799015221 168 144.76482730108395 1e+04 1246.1372158993884 1229 1085.7362047581294 ``` ## `pstats3(data_rows=29)` Return statistics of prime numbers up to `data_rows` which in default is equal `29`. This value can be changed from a function argument, but it can't be greater than that value. It can't be lower than 1 also. Rows statistics starts from 10 up to 10^29, and they are for prime numbers in these ranges. Time complexity: `O(n)`. **Examples** ```python >>> pstats3() <Returns all table> >>> pstats3(-2) Incorrect argument >>> pstats3(3) x π(x) PNT: Li(x) Li(x) - π(x) (Li(x) - π(x)) / π(x) [%] 1e+01 4 6.165599504787298 2.165600 54.139988 1e+02 25 30.12614158407963 5.126142 20.504566 1e+03 168 177.60965799015221 9.609658 5.720035 ``` # Usage: `azntprimality` module functions ## `is_prime_naive1(n)` Return if number is a prime. Time complexity: `O(n)`. **Examples** ```python >>> is_prime_naive1(7) True ``` ## `is_prime_naive2(n)` Return if number is a prime. This function is faster than `is_prime_naive1()` function. Time complexity: `O(n)`. **Examples** ```python >>> is_prime_naive2(5) True ``` ## `is_prime_sqrt(n)` Return if number is a prime. This function is faster than `is_prime_naive2()` function. Time complexity: `O(sqrt(n)))`. **Examples** ```python >>> is_prime_sqrt(101) True ``` ## `is_prime_sqrt_odd(n)` Return if number is a prime. This function is faster than `is_prime_sqrt()` function. Time complexity: `O(sqrt(n)))`. **Examples** ```python >>> is_prime_sqrt(23) True ``` ## `eratosthenes_sieve(n)` Return list of prime numbers from 2 up to `n` range. Time complexity: `O(n log n)`. **Examples** ```python >>> eratosthenes_sieve(100) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97] ``` # Usage: `azntplots` module functions ## `critical_strip()` Return a plot presenting critical strip and critical line of Riemann zeta. ## `riemann_zeta()` Return a plot presenting Riemann zeta function for critical line arguments.

نیازمندی

مقدار	نام
-	mpmath
-	matplotlib
-	numpy

زبان مورد نیاز

مقدار	نام
>=3.11	Python

نحوه نصب

نصب پکیج whl aznt-0.0.41:

pip install aznt-0.0.41.whl

نصب پکیج tar.gz aznt-0.0.41:

pip install aznt-0.0.41.tar.gz