An Elementary Method for Determining the Quadratic Character of -1, 2, 3 and 5 Modulo Primes

Arto Adili^1, , Adrian Naço² and Lorena Kelo¹

¹Department of Mathematics and Physics, “Fan S. Noli” University, Korçë, Albania

²Department of Mathematics Engineering, Polytechnic University, Tirana, Albania

Cite this paper:
Arto Adili, Adrian Naço and Lorena Kelo. An Elementary Method for Determining the Quadratic Character of -1, 2, 3 and 5 Modulo Primes. Turkish Journal of Analysis and Number Theory. 2026; 14(1):23-28. doi: 10.12691/tjant-14-1-3

Abstract

We provide elementary proofs of the classical quadratic-residue criteria for -1, 2, 3, and 5 modulo odd primes. In addition, we determine the quadratic character of the integers a and b in the prime representations p=a²+b² and p=a²+2b². These results further illustrate the relationship between quadratic residues and binary quadratic forms.

Keywords:
Quadratic character quadratic residue quadratic non-residue odd prime

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