fourresidues

Fourresidues is a term used in number theory to denote the fourth power residues modulo n: the set of integers a modulo n for which there exists an integer x with x^4 ≡ a (mod n). These residues form the image of the map x ↦ x^4 on the ring Z/nZ.

For a prime p, the structure is well understood. The multiplicative group of units modulo p has order p−1 and is cyclic. The nonzero fourth powers form a subgroup whose size is (p−1)/gcd(4, p−1). Including zero (since x ≡ 0 gives 0^4 ≡ 0), the total number of fourth residues modulo p is (p−1)/gcd(4, p−1) + 1. In particular, if p ≡ 1 mod 4, there are (p−1)/4 + 1 residues; if p ≡ 3 mod 4, there are (p−1)/2 + 1 residues. A fourth power is always a quadratic residue, but not every quadratic residue is necessarily a fourth power.

Beyond primes, the set of fourth residues modulo a composite n can be analyzed via the Chinese Remainder Theorem. If n factors into prime powers, the fourth residues modulo n correspond to tuples of fourth residues modulo each prime power, and the total count multiplies accordingly.

Computationally, to test whether a is a fourth residue modulo an odd prime p, one can check a^((p−1)/gcd(4, p−1)) ≡ 1 (mod p); this uses the cyclic structure of F_p^. For composite moduli, checks are performed modulo each prime power factor.

See also quadratic residues, higher power residues, modular arithmetic, and quartic reciprocity.

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