Epäjäännös

Epäjäännös, often translated as "non-residue," is a term used in certain mathematical contexts, particularly in number theory. It refers to an integer that cannot be expressed as a specific type of remainder or residue after division by a certain number. The most common context for this term is quadratic non-residues.

A quadratic residue modulo n is an integer 'a' such that the congruence x^2 ≡ a (mod n) has a solution for x. Conversely, a quadratic non-residue modulo n is an integer 'a' for which this congruence has no solution. For example, modulo 5, the squares are 1^2 ≡ 1, 2^2 ≡ 4, 3^2 ≡ 9 ≡ 4, and 4^2 ≡ 16 ≡ 1. Thus, 1 and 4 are quadratic residues modulo 5. The numbers 2 and 3 are not squares modulo 5, making them quadratic non-residues.

The concept of non-residues extends beyond quadratic congruences. More generally, if we consider a set of possible remainders when dividing by a number 'n', an epäjäännös would be an integer that does not belong to this set of allowed remainders according to a specific rule or property. The study of residues and non-residues is fundamental in areas like cryptography and the analysis of integer properties.