Neliöresiduaa

Neliöresiduaa is a mathematical concept in number theory dealing with the remainders of squares when divided by a given integer. Specifically, an integer 'a' is called a quadratic residue modulo 'n' if there exists an integer 'x' such that x^2 is congruent to 'a' modulo 'n'. This is written as x^2 ≡ a (mod n). If no such integer 'x' exists, then 'a' is called a quadratic non-residue modulo 'n'. The integer 'n' is called the modulus.

The study of quadratic residues is closely tied to the solvability of quadratic congruences. For example, when considering modulo a prime number 'p', a number 'a' is a quadratic residue modulo 'p' if the congruence x^2 ≡ a (mod p) has a solution. Euler's criterion provides a way to determine if an integer 'a' is a quadratic residue modulo an odd prime 'p'. It states that if 'p' does not divide 'a', then 'a' is a quadratic residue modulo 'p' if and only if a^((p-1)/2) ≡ 1 (mod p), and 'a' is a quadratic non-residue modulo 'p' if and only if a^((p-1)/2) ≡ -1 (mod p).

The concept is fundamental in understanding the properties of integers and their relationships under modular arithmetic. It has applications in various fields, including cryptography and the theory of numbers. The behavior of quadratic residues can be quite complex, especially when the modulus 'n' is composite. The Legendre symbol and the Jacobi symbol are important notations used to represent whether an integer is a quadratic residue modulo a prime or an odd number, respectively.