Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue-Nagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 1.

In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pad'e approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques. 1.

Wiles ’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. While celebrating Wiles ’ achievement, one also feels a twinge of regret at Fermat’s demise. Is the Holy Grail of number theorists to become a mere footnote in the history books? Hoping to keep some of the spirit of Fermat alive, I would like to discuss the generalized Fermat equation x p + y q = z r, (1) where p, q and r are fixed exponents. As in the case of Fermat’s Last Theorem, one is interested in integer solutions (x, y, z), which are non-trivial in the sense that xyz ̸ = 0. One might expect the equation above to have no such solutions if the exponents p, q, and r are large enough. But observe that, if p = q is odd, and r = 2, then any solution to a p +b p = c (of which there is an abundant supply!) yields the solution (ac, bc, c p+1 2) to the equation xp + yp = z2. A similar construction works whenever the exponents p, q, and r are pairwise coprime. However, the solutions produced in this way are not very interesting: the integers x, y and z have a large common factor. ∗This is a transcription of the author’s Aisenstadt prize lecture given at the CRM in March 1997. It is a pleasure to thank Andrew Granville and Loïc Merel for stimulating collaborations related to the topics of this essay, as well as Dan Abramovich for many helpful conversations over the years. This research was supported by CICMA and by grants from the Sloan Foundation, NSERC and FCAR. 1 Accordingly, one calls a solution (x, y, z) to the generalized Fermat equation primitive if gcd(x, y, z) = 1. Main Question: What are the non-trivial primitive solutions to the generalized Fermat equation? In [DG], Andrew Granville and I made the following conjecture:

We prove that the equation A 4 +B 2 = C p has no solutions in coprime positive integers when p≥211. The main step is to show that, for all sufficiently large primes p, every Q-curve over an imaginary quadratic field K with a prime of bad reduction greater than 6 has a surjective mod p Galois representation. The bound on p depends on K and the degree of the isogeny between E and its Galois conjugate, but is independent of the choice of E. The proof of this theorem combines geometric arguments due to Mazur, Momose, Darmon, and Merel with an analytic estimate of the average special values of certain L-functions. 1

1 Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication. For any positive integer k, let φk be the Galois representation associated to the k-division points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which φl is not surjective. 1

Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three non-zero distinct n-th powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One cannot fail to notice that it is a variant of the Fermat-Wiles theorem. We would like to present the ideas which led H. Darmon and the author to the solution of Denes' problem in [13]. Many of them are those (due to Y. Hellegouarch, G. Frey, J.-P. Serre, B. Mazur, K. Ribet, A. Wiles, R. Taylor, ...) which led to the celebrated proof of Fermat's last theorem. Others originate in earlier work of Darmon (and Ribet). The proof of Fermat's last theorem

1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1

Abstract. We find the primitive integer solutions to x 2 + y 3 = z 7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods. 1.

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