关于不定方程x^2-3y^4=166
摘要
The study of the Diophantine equation x^2-Dy^4=N (D and N are the given integers, D＞0 and D is non-square) has caused some authors interests, such as Cohn, Tzanakis, LI Jin-xiang, LIN LI-juan. Cohn has proven some conclusions. For example: N(5, 44)=1, (x, y)=(7, 1); N(5, 11)=2, (x, y)=(4, 1), (56, 5); N(5, -44)=3, (x, y)=(6, 2), (19, 3), (181, 9). Tzanakis has proven some conclusions while y≡0(mod8).For example: N(2, 17)=0, N(2, 41)=0, N(8, 17)=0, N(2, 97)=0. LI Jin-xiang has proven one conclusion: N(3, 46)=2, (x, y)=(7, 1), (17, 3). LIN LI-juan has also proven one conclusion: N(3, 22)=2, (x, y)=(5, 1), (85,7). But this Diophantine equation x^2-3y^4=166 still has not been solved until now. In this paper the author has proved that the Diophantine equation x^2-3y^4=166 has only positive integral solutions (x, y)=(13, 1), (193, 13) with the primary methods of recursive sequence, quadratic remainder and congruence.