Geodesic
On the discriminant of a quadratic field with intermediate fractions of negative norm and the decomposability of its representing polynomial
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 319-331 Cet article a éte moissonné depuis la source Math-Net.Ru

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The work is devoted to the study of Diophantine equation $x^2-y^2(p^{2}-4q)=4t$, where $p=l+u(k^2-1)(l(k^2-1)-2k)$, $q=u(lk^3-2k^2-kl+1)+km+1$, $l=k+m(k^{2}-1)$, numbers $k,m,u$ are nonnegative integers, number $k$ is odd, and the right hand side $4t$ of the equation is sufficiently small positive integer. We give a complete description of solutions of the Diophantine equation.
Mots-clés : diophantine equation, integer solutions, diophantine approximations.
Keywords: generalized Pell's equation, quadratic fields, unit group 
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A. A. Korobov; O. A. Korobov. On the discriminant of a quadratic field with intermediate fractions of negative norm and the decomposability of its representing polynomial. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 319-331. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a4/

[1] F. Luca, C.F. Osgood, P.G. Walsh, “Diophantine approximations and a problem from the 1988 IMO”, Rocky Mt. J. Math., 36:2 (2006), 637–648 | DOI | MR | Zbl

[2] A.A. Korobov, “On solvability conditions of one family of Pell's equations”, Mat. Zamet. YAGU, 20:1 (2013), 44–49 | Zbl

[3] A.A. Bukhshtab, Number Theory, Prosveschenie, M., 1966 | MR | Zbl

[4] G.J. Janusz, Algebraic number fields, Graduate Studies in Mathematics, 7, AMS, Providence, RI, 1996 | MR | Zbl