Solutions of congruence equations via an imprimitive action of some modular subgroups

S. Öztürk, Y. Kesicioğlu, H. Şengül

Let N be a positive integer and N = p₁^α₁ . p₂^α₂ ... p_r^α_r the prime power decomposition of N where p_i ≡ 1 (mod 4), for all 1≤i≤r. There, without the Girard-Fermat Theorem by no means, are integers a and x such that a²+x² ≡ 0 (mod N) by using a special imprimitive action of some modular subgroups. Finally we put a conjecture at the end of the paper.

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 16,  supplement issue 3 (2023), pp. 139-143