Neliöresidut

Neliöresidut, or square residues in English, are integers that are congruent to perfect squares modulo some given integer n. In mathematical terms, a number a is a quadratic residue modulo n if there exists an integer x such that x² ≡ a (mod n). This concept is fundamental in number theory and has applications in cryptography and primality testing.

The study of quadratic residues dates back to the works of Euler and Gauss, who established many of their basic properties. The Legendre symbol and Jacobi symbol are important notations used to determine whether a number is a quadratic residue modulo a prime or composite number, respectively.

For example, modulo 7, the quadratic residues are 0, 1, 2, and 4, as these are the possible squares of integers when divided by 7. The remaining numbers, 3, 5, and 6, are quadratic non-residues modulo 7.

Quadratic residues exhibit interesting distribution patterns that have been extensively researched. Their properties form the foundation for many modern cryptographic systems and continue to be an active area of mathematical investigation. The study of these residues provides insights into the deeper structure of number systems and modular arithmetic.