Let C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let $n \ge 2$. We say that an algebraic point $P \in C(\overline {K})$ has degree n if the extension $K(P)/K$ has degree n. By the Galois group of P we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$. We say that P is primitive if its Galois group is primitive as a subgroup of $S_n$. We prove the following ‘single source’ theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$. Suppose that either J is simple or that $J(K)$ is finite. Suppose C has infinitely many primitive degree n points. Then there is a degree n morphism $\varphi : C \rightarrow \mathbb {P}^1$ such that all but finitely many primitive degree n points correspond to fibres $\varphi ^{-1}(\alpha )$ with $\alpha \in \mathbb {P}^1(K)$.

We prove, moreover, under the same hypotheses, that if C has infinitely many degree n points with Galois group $S_n$ or $A_n$, then C has only finitely many degree n points of any other primitive Galois group.

Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation

$$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$

$abc=0$

$K$

$K$

$K=\mathbb{Q}(\sqrt{d})$

$d\geqslant 2$

${\textstyle \frac{5}{6}}$

$1$

Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.

In this paper, we determine all the factorials that are a sum of at most three Fibonacci numbers.

We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation

$${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$

in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$ . Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

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 x2 + D = yn, x, y integers, $n\geq 3$, for D in the range $1 \leq D \leq 100$.

Let $X$ be the Shimura curve corresponding to the quaternion algebra over $\Q$ ramified only at 3 and 13. B. Jordan showed that $X_{\Q(\sqrt{-13})}$ is a counterexample to the Hasse principle. Using an equation of $X$ found by A. Kurihara, it is shown here, by elementary means, that $X$ has no $\Q(\sqrt{-13})$-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.

Let k be a perfect field, X a smooth curve over k, and denote by Xc the subset of closed points of X. We show that for any non-constant element f of the function field k (X) there exists a natural homomorphismWhere

We explain how this generalises the usual results on descents on Jacobians and Picard groups of curves.