An integer r (or its congruence class modulo n) that is not congruent to any square modulo n; equivalently r is outside the image of the squaring map x ↦ x^2 on Z/nZ.

Principle

Nonresidues are defined negatively by failure of solvability: the principle is classification of classes by whether a quadratic congruence x^2 ≡ r (mod n) admits a solution, separating classes into residues and nonresidues.

Demonstration

Modulo 7 the class of 3 is a quadratic nonresidue because no x in {0,…,6} satisfies x^2 ≡ 3 (mod 7); by contrast 2 and 4 are residues as shown by explicit squares.

Misapplication

Calling a number a nonresidue without specifying modulus, or transferring parity-count heuristics from prime moduli directly to composite moduli where distribution of nonresidues is more intricate.

Consequence

Recognizing nonresidues identifies classes where quadratic congruences lack solutions, informs reciprocity laws and cryptographic primitives that rely on hardness of distinguishing residues from nonresidues in some settings.

Reversal

The reverse notion is a quadratic residue; swapping classification changes whether quadratic equations are solvable and alters multiplicative subgroup membership in (Z/nZ)^×.

Boundary

Depends on modulus n: for odd prime p about half of nonzero classes are nonresidues (counting conventions aside); for composite n zero divisors and Chinese remainder decomposition change existence and counting.

Semantic Tension

Tension appears between the algorithmic task 'test whether a given integer is a nonresidue' and the abstract classification; also between statistical heuristics and exact algebraic criteria provided by symbols like Legendre or Jacobi.

Synthesis

A quadratic nonresidue modulo n is a congruence class excluded from the image of squaring; together with residues it partitions classes by solvability of x^2 ≡ r and plays a central role in reciprocity, algorithms, and cryptographic constructions.