CC BY
11
Journal of Siberian Federal University. Mathematics & Physics 2019, 12(1), 130
УДК 511.52
Errata to Our Article
Nikolay N. Osipov* Bella V. Gulnovat
Institute of Space and Information Technology Siberian Federal University Kirensky, 26, Krasnoyarsk, 660074
Russia
In our paper [1] the estimate (21) for the solutions of the equation (3) is correct with the exception of at most two easily computable integer solutions (x,y). Namely, in the case when the discriminant Д > 0 is a perfect square, the numbers k± = —а5к± with
—B ± ^Д = 2C
may be integer so the coefficient a = Ck2 — Ba5k + Aa2 may vanishes at k = k±. In this case the general formula (10) for x is not valid and we compute the solution (x, y) as follows:
Y
a^k — Ca2 — a^a^ + a2a^a^ kx + a@
x ^--^ -, y ^ ,
¡3 (2Ca6 — a2a^)k + a^ag — Ba^a^ a5 '
where k = k±. Geometrically, this exceptional solution (x,y) corresponds to such rational point on the curve (3) that can be obtained as the third intersection point of the line passing through two rational points [0, —a6/a5,1], [1, k±, 0] (in homogeneous coordinates) and the curve. In particular, for the equation
x(y2 — x2) + Hx + y +1 = 0 from Example 10 we obtain (for even H) precisely two exceptional solutions
(x, y) = (H/2 + 1, H/2), (x, y) = (-H/2, H/2 — 1) and the proposed estimate \x\ < C8\H\l/2 is correct only for non-exceptional solutions.
References
[1] N.N.Osipov, B.V.Gulnova, An algorithmic implementation of Runge's method for cubic diophantine equations, J. Sib. Fed. Univ. Math. Phys., 11(2018), no. 2, 137-147.
Опечатки в нашей статье
Николай Н. Осипов Белла В. Гульнова
* nnosipov@rambler.ru tecureuil66b@gmail.com
