math.integer — integer-specific mathematics functions

Added in version 3.15.

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This module provides access to the mathematical functions defined for integer arguments. These functions accept integers and objects that implement the __index__() method which is used to convert the object to an integer number.

The following functions are provided by this module. Unless stated otherwise below, all return values are computed exactly and are integers.

math.integer.comb(n, k, /)

Return the number of ways to choose k items from n items without repetition and without order.

Evaluates to n! / (k! * (n - k)!) when k <= n and evaluates to zero when k > n.

Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of (1 + x)ⁿ.

Raises ValueError if either of the arguments are negative.

math.integer.factorial(n, /)

Return factorial of the nonnegative integer n.

math.integer.gcd(*integers)

Return the greatest common divisor of the specified integer arguments. If any of the arguments is nonzero, then the returned value is the largest positive integer that is a divisor of all arguments. If all arguments are zero, then the returned value is 0. gcd() without arguments returns 0.

math.integer.isprime(n, /)

Return True if n is a prime number, False otherwise.

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. Negative numbers, 0 and 1 are not prime.

The argument must be less than 2⁶⁴; raises OverflowError otherwise.

Added in version 3.16.0a0 (unreleased).

math.integer.isqrt(n, /)

Return the integer square root of the nonnegative integer n. This is the floor of the exact square root of n, or equivalently the greatest integer a such that a² ≤ n.

For some applications, it may be more convenient to have the least integer a such that n ≤ a², or in other words the ceiling of the exact square root of n. For positive n, this can be computed using a = 1 + isqrt(n - 1).

math.integer.lcm(*integers)

Return the least common multiple of the specified integer arguments. If all arguments are nonzero, then the returned value is the smallest positive integer that is a multiple of all arguments. If any of the arguments is zero, then the returned value is 0. lcm() without arguments returns 1.

math.integer.perm(n, k=None, /)

Return the number of ways to choose k items from n items without repetition and with order.

Evaluates to n! / (n - k)! when k <= n and evaluates to zero when k > n.

If k is not specified or is None, then k defaults to n and the function returns n!.

Raises ValueError if either of the arguments are negative.

math.integer.primes(start=2, stop=None)

Return an iterator of the prime numbers p with start <= p < stop, in increasing order. If stop is None (the default), the iteration does not stop.

Roughly equivalent to (p for p in itertools.count(start) if isprime(p)) for the unbounded form.

The bounds must be less than 2⁶⁴; raises OverflowError otherwise. Unbounded iteration raises OverflowError if the candidates reach that limit.

Added in version 3.16.0a0 (unreleased).