We investigate solutions to the matrix Diophantine equation $ X^n + Y^n = n!Z^n $ within specific classes of $2 \times 2$ integer matrices. Firstly, we introduce a class of $2\times 2$ matrices $G_2(d)$ and show that $X^n + Y^n = n!Z^n$ admits a solution in $G_2(d)$ if and only if the classical Diophantine equation $x^n + y^n = n!z^n$ has a solution in $\mathbb {Q}(\sqrt {d})$. We further present a constructive theorem for generating matrix solutions over a general number field K. Finally, we introduce a determinant-fixed class of matrices $G_2(d, l)$, and prove that the Fermat-type equation $X^n+Y^n=2Z^n$ has only the trivial solution in $G_2(d, l)$ whenever $n>2$.

Let $a\ge 1$. Denote by $N_a(n)$ the number of solutions of

$$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$

$a>1,$

$\sum \limits _{a\le n\le x}N_a(n)$

$x\to \infty $

$\sum \limits _{a\le n\le x}nN_a(n)$

$\sum \limits _{a\le n\le x}\frac {1}{n}N_a(n)$

$\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d),$

$\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d)\,\tau \!\left (\frac {n}{d}\right )$

$\varepsilon>0$

$ N_a(n)<n^{1+\varepsilon } $

Let $\{u_n\}_n$ be a nondegenerate linear recurrence sequence of integers with Binet’s formula given by ${u_n= \sum _{i=1}^{m} P_i(n)\alpha _i^n.}$ Assume $\max _i \vert \alpha _i \vert>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon>0$, there is an effectively computable constant $C(\epsilon )$ such that if $ \vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, then $n<C(\epsilon )$. Using results of Schmidt and Evertse, a complete noneffective (qualitative) proof of this conjecture was given by Fuchs and Heintze [‘On the growth of linear recurrences in function fields’, Bull. Aust. Math. Soc. 104(1) (2021), 11–20] and, independently, by Karimov et al. [‘The power of positivity’, Proc. LICS 23 (2023), 1–11]. In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

Ishitsuka et al. [‘Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic’, Int. J. Number Theory 16(4) (2020), 881–905] found all points on the Fermat quartic ${F_4\colon x^4+y^4=z^4}$ over quadratic extensions of ${\mathbb {Q}}(\zeta _8)$, where $\zeta _8$ is the eighth primitive root of unity $e^{i\pi /4}$. Using Mordell’s technique, we give an alternative proof for the result of Ishitsuka et al. and extend it to the rational function field ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$.

We investigate the equation $D=x^4-y^4$ in field extensions. As an application, for a prime number p, we find solutions to $p=x^4-y^4$ if $p\equiv 11$ (mod 16) and $p^3=x^4-y^4$ if $p\equiv 3$ (mod 16) in all cubic extensions of $\mathbb{Q}(i)$.

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$, where $1\le k_1<k_2<\cdots <k_r=k$. Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$, we show that there is a paucity of nondiagonal solutions to the Diophantine system $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\mathbb {Q}(\sqrt {-d})$, where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.

We show that the system of equations

$$ \begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$

has appreciably fewer solutions in the subcritical range $s < \tfrac 12k(k+1)$ than its homogeneous counterpart, provided that $a_{\ell } \neq 0$ for some $\ell \leqslant k-1$. Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

We prove that if $s\ge 2$ is a fixed integer, then the equation $ns^n+1=(b^m-1)/(b-1)$ has only finitely many positive integer solutions $(n,b,m)$ with $b\ge 2$ and $m\ge 3$. When $s=2$, it has no solution.

Let $\mathcal {P}_s(n)$ denote the nth s-gonal number. We consider the equation

$$ \begin{align*}\mathcal{P}_s(n) = y^m \end{align*} $$

for integers $n,s,y$ and m. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number greater than 1 expressible as a perfect mth power with $m>1$ is $\mathcal {P}_{10}(3) = 3^3$.

Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every $n\geq 2$ in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.

In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

We sharpen earlier work of Dabrowski on near-perfect power values of the quartic form $x^{4}-y^{4}$, through appeal to Frey curves of various signatures and related techniques.

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

Weinvestigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.