On a conjecture concerning the generalized Ramanujan-Nagell equation.(English)Zbl 1501.11050
Let \(c,d\) be fixed coprime positive integers with \(\min \{c,d\} > 1\). A class of polynomial-exponential Diophantine equations of the form \[x^2+d^y=c^z,\,\,x,y,z\in\mathbb{Z}^{+}\tag{1}\]is usually called the generalized Ramanujan-Nagell equation. It has a long history and rich content (see [M. Le andG. Soydan, Surv. Math. Appl. 15, 473–523 (2020;Zbl 1482.11054)]). In 2014, N. Terai discussed the solution of (1) in the case \(d=2c-1\). He proposed the following conjecture:
Conjecture 1. For any \(c\) with \(c>1\), the equation \[x^2+(2c-1)^y=c^z,\,\,x,y,z\in\mathbb{Z}^{+}\tag{2}\]has only one solution \((x,y,z)=(c-1,1,2)\).
Although the above conjecture has been verified in some special cases, it is still a far from solved problem in the general case.
Given any fixed positive integer \(a\), there exist unique positive integers \(a_1,a_2\) such that \(a=a_1a_2^2\) and \(a_1\) is square free. Then, by notation in the paper \(a_1\) is called the quadratfrei of \(a\), and denoted by \(q(a)\)
In this paper, the authors derive some criteria for Conjecture 1 to be true as follows.
Conjecture 1 is true if one of the following conditions is satisfied:
If \(N\) is large enough, then \(f(N)/N>0.78\).
Conjecture 1. For any \(c\) with \(c>1\), the equation \[x^2+(2c-1)^y=c^z,\,\,x,y,z\in\mathbb{Z}^{+}\tag{2}\]has only one solution \((x,y,z)=(c-1,1,2)\).
Although the above conjecture has been verified in some special cases, it is still a far from solved problem in the general case.
Given any fixed positive integer \(a\), there exist unique positive integers \(a_1,a_2\) such that \(a=a_1a_2^2\) and \(a_1\) is square free. Then, by notation in the paper \(a_1\) is called the quadratfrei of \(a\), and denoted by \(q(a)\)
In this paper, the authors derive some criteria for Conjecture 1 to be true as follows.
Conjecture 1 is true if one of the following conditions is satisfied:
- (i)
- \(c\equiv 3 \pmod{4}\) and \(q(c-1)>2.8\log c\).
- (ii)
- \(c\equiv 0 \pmod{4}\), \(2c-1\) has a prime divisor \(p\) with \(p\equiv \pm 3 \pmod{8}\) and \(q(c-1)>4.8 \log c\)
- (iii)
- \(c\equiv 1 \pmod{4}\) and \[q(c-1)>\max \{10^5,2.8\log c,\dfrac{2}{\pi}\sqrt{2c-1}(2+2\log 2+\log(2c-1)\} \]
If \(N\) is large enough, then \(f(N)/N>0.78\).
Reviewer: Gökhan Soydan (Bursa)
MSC:
| 11D61 | Exponential Diophantine equations |
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11J86 | Linear forms in logarithms; Baker’s method |
