Classical and modular approaches to exponential Diophantine equations. II: The Lebesgue-Nagell equation.

*(English)*Zbl 1128.11013In the first of two papers [Ann. Math. (2) 163, No. 3, 969–1018 (2006; Zbl 1113.11021)], the three authors proved that there are no perfect powers – except a small set of previously known exceptions – in two classical quadratic recurrent sequences: the Fibonacci sequence \(F_{n+2} = F_{n+1} + F_{n}, F_0 = F_1 = 1\) and the particular Lucas sequence \(L_{n+2} = L_{n+1} + L_n, L_0 = 2, L_1 = 1\). For this they applied an astute combination of methods based on Baker’s forms in logarithms, modular forms and Ribet descent together with elementary congruence equations.

In the paper under review, the same arsenal of methods is brought to work for the investigation of the exponential Diophantine equation \[ x^2 + D = y^n, \quad x, y \in \mathbb Z \quad \text{ and } \;n \geq 3 ,\tag{rn} \] which the authors denote by generalized Ramanujan-Nagell equation. The equation is naturally reduced to the case when \(n = p\) is a prime, by noting that a solution \((x, y, n)\) with \(p | n\) leads to a solution \((x, y^{n/p}, p)\). The paper gives the complete set of solutions for values \(1 \leq D \leq 100\), and herewith also solves an old standing problem raised by Ramanujan, related to the particular case \(D = 7\). The list of solutions of (rn) is displayed at the end of the paper and the proofs accumulate in the direction of showing that the list is complete. It begins with a useful overview of previous results related to (rn). In Section 3, a preliminary step uses reduction of Thue equations, yielding some lower bounds on \(p\).

It is important to remove possible common factors of the three parameters in (rn), and this leads to a case distinction based on the square free part \(d_2 | D\) and the equation \[ t^2 + d_2 = e s^p, \quad \text{with} \quad \text{rad}(e) = \text{rad}(D/d_1) \] This equation is investigated in section 6 by using level lowering techniques. Thus one obtains a restricted list of congruences for \(D, p\).

The next step consists in eliminating exponents, which is first done by using congruences gained with the level lowering methods. This also reduces the possible values of \(D\) to a list of \(19\) and yields a (relatively small) lower bound for \(| y |\), while \(p \geq 10^8\).

The linear forms in logarithms are used in general for exponential equations in order to provide upper bounds on the exponents of possible equations. A theorem of Matveev, together with specific machinery, that was developed by Mignotte in dealing with special cases of (rn) and extended in the present paper with a new estimate, eventually yield the upper bound \(p < 4.2 \times 10^8\). The remaining gap is finally investigated, by a combination of all the methods developed so far, thus completing the proof of the main result of the paper: equation (rn) has, for \(1 \leq D \leq 100\), exactly the solutions displayed in the paper.

In the paper under review, the same arsenal of methods is brought to work for the investigation of the exponential Diophantine equation \[ x^2 + D = y^n, \quad x, y \in \mathbb Z \quad \text{ and } \;n \geq 3 ,\tag{rn} \] which the authors denote by generalized Ramanujan-Nagell equation. The equation is naturally reduced to the case when \(n = p\) is a prime, by noting that a solution \((x, y, n)\) with \(p | n\) leads to a solution \((x, y^{n/p}, p)\). The paper gives the complete set of solutions for values \(1 \leq D \leq 100\), and herewith also solves an old standing problem raised by Ramanujan, related to the particular case \(D = 7\). The list of solutions of (rn) is displayed at the end of the paper and the proofs accumulate in the direction of showing that the list is complete. It begins with a useful overview of previous results related to (rn). In Section 3, a preliminary step uses reduction of Thue equations, yielding some lower bounds on \(p\).

It is important to remove possible common factors of the three parameters in (rn), and this leads to a case distinction based on the square free part \(d_2 | D\) and the equation \[ t^2 + d_2 = e s^p, \quad \text{with} \quad \text{rad}(e) = \text{rad}(D/d_1) \] This equation is investigated in section 6 by using level lowering techniques. Thus one obtains a restricted list of congruences for \(D, p\).

The next step consists in eliminating exponents, which is first done by using congruences gained with the level lowering methods. This also reduces the possible values of \(D\) to a list of \(19\) and yields a (relatively small) lower bound for \(| y |\), while \(p \geq 10^8\).

The linear forms in logarithms are used in general for exponential equations in order to provide upper bounds on the exponents of possible equations. A theorem of Matveev, together with specific machinery, that was developed by Mignotte in dealing with special cases of (rn) and extended in the present paper with a new estimate, eventually yield the upper bound \(p < 4.2 \times 10^8\). The remaining gap is finally investigated, by a combination of all the methods developed so far, thus completing the proof of the main result of the paper: equation (rn) has, for \(1 \leq D \leq 100\), exactly the solutions displayed in the paper.

Reviewer: Preda Mihailescu (GĂ¶ttingen)