Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
generalized Ramanujan-Nagell equation; number of solution; upper bound
generalized Ramanujan-Nagell equation; number of solution; upper bound
Summary:
Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D, p)$, and also prove that if the equation $U^2-DV^2=-1$ has integer solutions $(U, V)$, the least solution $(u_1, v_1)$ of the equation $u^2-pv^2=1$ satisfies $p\nmid v_1$, and $D>C(p)$, where $C(p)$ is an effectively computable constant only depending on $p$, then the equation $x^2-D=p^n$ has at most two positive integer solutions $(x, n)$. In particular, we have $C(3)=10^7$.
References:
[1] Bauer, M., Bennett, M. A.: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209-270. DOI 10.1023/A:1015779301077 | MR 1908198 | Zbl 1010.11020
[2] Beukers, F.: On the generalized Ramanujan-Nagell equation II. Acta Arith. 39 (1981), 113-123. DOI 10.4064/aa-39-2-113-123 | MR 0639621 | Zbl 0377.10012
[3] Le, M. H.: On the generalized Ramanujan-Nagell equation $x^2-D=p^n$. Acta Arith. 58 (1991), 289-298. DOI 10.4064/aa-58-3-289-298 | MR 1121088
[4] Le, M. H.: On the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$. Publ. Math. Debrecen. 45 (1994), 239-254. MR 1315938
[5] Le, M. H.: Upper bounds for class numbers of real quadratic fields. Acta Arith. 68 (1994), 141-144. DOI 10.4064/aa-68-2-141-144 | MR 1305196 | Zbl 0816.11055
[6] Mordell, L. J.: Diophantine Equations. London, Academic Press. (1969). MR 0249355 | Zbl 0188.34503
[7] Siegel, C. L.: Approximation algebraischer Zahlen. Diss. Göttingen, Math. Zeitschr. 10 (1921), 173-213 German. DOI 10.1007/BF01211608 | MR 1544471
Search
Partner of
