jäännösneliöiden

Jäännösneliöt, or quadratic residues in English, are a fundamental concept in number theory. An integer 'a' is considered a quadratic residue modulo 'n' if there exists an integer 'x' such that x^2 is congruent to 'a' modulo 'n'. This means that when x^2 is divided by 'n', the remainder is 'a'. Conversely, if no such 'x' exists, then 'a' is called a quadratic non-residue modulo 'n'.

The study of quadratic residues is primarily concerned with prime moduli. For a prime modulus 'p', an integer 'a' not divisible by 'p' is a quadratic residue if the congruence x^2 ≡ a (mod p) has a solution. Otherwise, it is a quadratic non-residue. The Legendre symbol, denoted as (a/p), is a convenient notation to indicate whether 'a' is a quadratic residue modulo 'p'. It is equal to 1 if 'a' is a quadratic residue and -1 if 'a' is a quadratic non-residue. If 'a' is divisible by 'p', the Legendre symbol is 0.

The distribution of quadratic residues and non-residues is remarkably regular. For an odd prime 'p', exactly half of the non-zero residues modulo 'p' are quadratic residues, and the other half are quadratic non-residues. This property has significant implications in various areas of mathematics, including cryptography and coding theory. Understanding jäännösneliöt is crucial for solving quadratic congruences and for analyzing the properties of number systems.