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In this article, we use the elementary methods and the estimate for character sums to study a problem related to quadratic residues and the Pythagorean triples, and prove the following result. Let $ p $ be an odd prime large enough. Then for any positive number $ 0 < \epsilon < 1 $, there must exist three quadratic residues $ x, \ y $ and $ z $ modulo $ p $ with $ 1\leq x, \ y, \ z\leq p^{1+\epsilon} $ such that the equation $ x^2+y^2 = z^2 $.

Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai [6] conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).