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( Elif Kizildere ),( Gökhan Soydan ) 호남수학회 2020 호남수학학술지 Vol.42 No.1
Let p be a prime number with p > 3, p ≡ 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn<sup>2</sup> - 1)<sup>x</sup> + (p(p - 5)n<sup>2</sup> + 1)<sup>y</sup> = (pn)<sup>z</sup> has only the positive integer solution (x; y; z) = (1; 1; 2) where pn ≡ ±1 (mod 5). As an another result, we show that the Diophantine equation (35n<sup>2</sup> - 1)<sup>x</sup> + (14n<sup>2</sup> + 1)<sup>y</sup> = (7n)<sup>z</sup> has only the positive integer solution (x, y, z) = (1, 1, 2) where n ≡ ±3 (mod 5) or 5 | n. On the proofs, we use the properties of Jacobi symbol and Baker's method.
On the additive structure of the set of quadratic residues modulo p
G. Soydan,N. Y. Ikikardes,M. Demirci,I. Cang?l 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.14 No.2
It is well-known that the set of quadratic residues modulo p forms multiplicative group. But except for very special cases, there is no result on the sum of quadratic residues or non-residues. Here we study three different sums; the sum of two quadratic residues, the sum of two quadratic non-residues and the sum of a quadratic residue with a non-residue. Bachet elliptic curves are the curves of the form y2 = x3 + a3 where a is a fixed element of the underlying field. Studying the rational points on these curves depends on our knowledge of the sum of a variable x3 and a fixed value a. Therefore we study such sums as well.