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Hardy G. H., Wright E. M. (1959): An Introduction to the Theory of Numbers, Fourth Edition, Oxford University Press (The. 85, pg. 69).

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Article

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

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

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


Turkish Journal of Analysis and Number Theory. 2026, Vol. 14 No. 1, 23-28
DOI: 10.12691/tjant-14-1-3
Copyright © 2026 Science and Education Publishing

Cite this paper:
Arto Adili, Adrian Naço, 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.

Correspondence to: Arto  Adili, Department of Mathematics and Physics, “Fan S. Noli” University, Korçë, Albania. Email: aadili@unkorce.edu.al

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.

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