Let s be a fixed positive integer constant and let $\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set ${\mathcal M}\subseteq \mathbb {F}_p^*$ of size $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer $H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$

When $s=1$, we show that, for almost all primes p, if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$

We show that for $5/6$-th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt [3]{p})$. We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt {D}, \sqrt [3]{p})$ for every $D \in S$ and for every prime $p \equiv 2, 5\ \pmod 9$. We use the CM elliptic curves $y^2=x^3-432 D^2$ associated with the cube-sum problem, with D varying in suitable congruence class, in our proof.

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.

For an integer $k \geq 2$, let $P_{n}^{(k)}$ be the k-generalised Pell sequence, which starts with $0, \ldots ,0,1$ (k terms), and each term thereafter is given by the recurrence $P_{n}^{(k)} = 2 P_{n-1}^{(k)} +P_{n-2}^{(k)} +\cdots +P_{n-k}^{(k)}$. We search for perfect powers, which are sums or differences of two k-generalised Pell numbers.

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.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

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 ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$.

Let n be a nonzero integer. A set S of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. We show the contribution of intermediate elements is $O(\log \log |n|)$, improving a result by Dujella [‘Bounds for the size of sets with the property $D(n)$’, Glas. Mat. Ser. III 39(59)(2) (2004), 199–205]. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$, improving the best known upper bound on $M_n$ by Becker and Murty [‘Diophantine m-tuples with the property $D(n)$’, Glas. Mat. Ser. III 54(74)(1) (2019), 65–75].

We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let $B_1,\dots ,B_r\in \mathrm {GL}_m(K)$ be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of $\mathrm {GL}_m(K)$ not passing through zero. Then there exist only finitely many elements of $\mathrm {GL}_m(K)$ of the form $B_1^{n_1}\cdots B_r^{n_r}$ (as we vary $n_1,\dots ,n_r$ in $\mathbb {Z}$) lying on the subvariety V.

Let f be an $L^2$-normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that

$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

We establish analogous estimates in the setting of definite quaternion algebras.

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

A generalisation of the well-known Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$, $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n\ge 0$ is the k-generalised Pell sequence $\{P^{(k)}_n\}_{n\ge -(k-2)}$ whose first k terms are $0,\ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$. For the Pell sequence, the formula $P^2_n+P^2_{n+1}=P_{2n+1}$ holds for all $n\ge 0$. In this paper, we prove that the Diophantine equation

$$ \begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m \end{align*} $$

has no solution in positive integers $k, m$ and n with $n>1$ and $k\ge 3$.

We prove the Hasse principle for a smooth projective variety $X\subset \mathbb {P}^{n-1}_\mathbb {Q}$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\mathbb {Q}$.

We propose generating functions, $\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.