eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2026-07-14 14 1 23 28 10.12691/tjant-14-1-3 TJANT20261413 article Arto Adili aadili@unkorce.edu.al 1 Adrian Na?o 2 Lorena Kelo 2 Department of Mathematics and Physics, ˇ°Fan S. Noliˇ± University, Kor??, Albania Department of Mathematics Engineering, Polytechnic University, Tirana, Albania 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. https://pubs.sciepub.com/tjant/14/1/3/tjant-14-1-3.pdf Quadratic character quadratic residue quadratic non-residue odd prime