num_theory

Submodules

`arithmetic_progression`(a, d, n[, compute_sum, nth_term])	Generate terms of an arithmetic progression (AP), compute the nth term, or calculate the sum of the first n terms.
`is_prime`(n)	Check if a number is prime.
`prime_factorization`(n)	Compute the prime factorization of a given integer.
`get_primes`(→ list)	Returns the list of all prime numbers less than or equal to n.

num_theory.arithmetic_progression(a, d, n, compute_sum=False, nth_term=False)[source]

Generate terms of an arithmetic progression (AP), compute the nth term, or calculate the sum of the first n terms.

Parameters:

a (float) – The first term of the AP.
d (float) – The common difference between consecutive terms.
n (int) – The number of terms, the term to compute, or the term index.
compute_sum (bool, optional) – If True, computes the sum of the first n terms. Default is False.
nth_term (bool, optional) – If True, computes the nth term of the AP instead of generating terms. Default is False.

Returns:

If compute_sum and nth_term are both False, returns a list of the first n terms of the AP.
If nth_term is True, returns the nth term as a float.
If compute_sum is True, returns the sum of the first n terms as a float.
If both nth_term and compute_sum is True, it will return the nth term as a float.

Return type:

list or float

Examples

>>> arithmetic_progression(a=2, d=3, n=5)
[2, 5, 8, 11, 14]

>>> arithmetic_progression(a=2, d=3, n=5, compute_sum=True)
40.0

>>> arithmetic_progression(a=2, d=3, n=5, nth_term=True)
14

>>> arithmetic_progression(a=1, d=2, n=5, nth_term=True)
9

num_theory.is_prime(n)[source]

Check if a number is prime.

Parameters:: n (int) – The number to check for primality. Must be a positive integer greater than 1.
Returns:: Returns True if the number is prime, otherwise False.
Return type:: bool
Raises:: ValueError – If the input is not a positive integer greater than 1.

Examples

>>> is_prime(2)
True
>>> is_prime(4)
False
>>> is_prime(13)
True

num_theory.prime_factorization(n)[source]

Compute the prime factorization of a given integer.

Parameters:: n (int) – The integer to factorize. Must be a positive integer greater than 1.
Returns:: A list of tuples where each tuple represents a prime factor and its corresponding power. Example: For n=28, the output will be [(2, 2), (7, 1)], indicating 28 = 2^2 * 7^1.
Return type:: list of tuple
Raises:: ValueError – If the input n is not a positive integer greater than 1.

Examples

>>> prime_factorization(28)
[(2, 2), (7, 1)]

>>> prime_factorization(100)
[(2, 2), (5, 2)]

Notes

This function uses trial division to determine the prime factors.
For large values of n, consider optimized algorithms such as Pollard’s Rho or the Quadratic Sieve for better performance.

num_theory.get_primes(num: int) → list[source]

Returns the list of all prime numbers less than or equal to n.

Examples

>>> get_primes(14)
[2, 5, 7, 11, 13]

>>> get_primes(5)
[2, 5]

>>> get_primes(-2)
[]