Pnfunktsiooni

Pnfunktsiooni (Estonian for “p-function”) is a term used in some mathematical writings to denote a family of functions parameterized by a positive integer p, commonly representing a residue or remainder operation modulo p. In its typical form, pnfunktsiooni refers to the function f_p: N → {0, 1, ..., p−1} defined by f_p(n) = n mod p, i.e., the remainder when n is divided by p.

Formal definition and example: For a fixed p ≥ 2, f_p(n) equals the least nonnegative remainder of n upon division by p. For instance, with p = 5, the sequence f_5(n) as n runs through 0, 1, 2, 3, 4, 5, 6, ... is 0, 1, 2, 3, 4, 0, 1, 2, and so on. This simple mapping is widely used to study modular arithmetic and residue classes.

Properties: The pnfunktsiooni is periodic with period p, since f_p(n + p) = f_p(n). Its range is the set {0, 1, ..., p−1}, and as n increases, each residue appears infinitely often. For prime p, the residues distribute cyclically with a straightforward pattern, which makes the function a common teaching tool for illustrating congruences and modular reasoning.

Variations and usage: Some writers generalize the concept to related maps, such as composing the remainder with another modulus (e.g., f_p,q(n) = (n mod p) mod q) or considering the quotient floor(n/p) as a different but related p-dependent function. In programming and discrete mathematics, the modulo operation embodied by pnfunktsiooni is used in hashing, circular buffers, and cyclic algorithms.

History and terminology: The term pnfunktsiooni is most often encountered in Estonian-language texts. In English, the same idea is usually described as the modulo p remainder function or simply the p-modulus function. See also modulo operation and residue classes.