We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size $B$ with each component having no prime divisors below $B^{1/u}$, where $u$ equals $c_{0}n^{3/2}$, $n$ is the number of variables and $c_{0}$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $n^{3/2}$ to the logarithmic $(\log n)(\log \log n)^{-1}$. Our main new ingredients are the generalization of the Brüdern–Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport–Birch version of the circle method.

We shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

Let $c\geq 2$ be a positive integer. Terai [‘A note on the Diophantine equation $x^{2}+q^{m}=c^{n}$’, Bull. Aust. Math. Soc.90 (2014), 20–27] conjectured that the exponential Diophantine equation $x^{2}+(2c-1)^{m}=c^{n}$ has only the positive integer solution $(x,m,n)=(c-1,1,2)$. He proved his conjecture under various conditions on $c$ and $2c-1$. In this paper, we prove Terai’s conjecture under a wider range of conditions on $c$ and $2c-1$. In particular, we show that the conjecture is true if $c\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$ and $3\leq c\leq 499$.

We establish bounds for triple exponential sums with mixed exponential and linear terms. The method we use is by Shparlinski [‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160] together with a bound for the additive energy from Roche-Newton et al. [‘New sum-product type estimates over finite fields’, Adv. Math.293 (2016), 589–605].

For the superelliptic curves of the form

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

$y\neq 0$

$k\geqslant 3$

$\ell \geqslant 2,$

$i\in [2,k]\setminus \unicode[STIX]{x1D6FA}$

$\ell <\text{e}^{3^{k}}.$

$\unicode[STIX]{x1D6FA}$

$[p_{\unicode[STIX]{x1D703}},(k-p_{\unicode[STIX]{x1D703}}))$

$p_{\unicode[STIX]{x1D703}}$

$k/2$

$i=1$

Let $s\geqslant 3$ be a fixed positive integer and let $a_{1},\ldots ,a_{s}\in \mathbb{Z}$ be arbitrary. We show that, on average over $k$ , the density of numbers represented by the degree $k$ diagonal form

$$\begin{eqnarray}a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}\end{eqnarray}$$

$k$

Let $K$ be an algebraic number field. A cuboid is said to be $K$ -rational if its edges and face diagonals lie in $K$ . A $K$ -rational cuboid is said to be perfect if its body diagonal lies in $K$ . The existence of perfect $\mathbb{Q}$ -rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields $K$ such that a perfect $K$ -rational cuboid exists; and that, for every integer $n\geq 2$ , there is an algebraic number field $K$ of degree $n$ such that there exists a perfect $K$ -rational cuboid.

For any odd prime $\ell$ , let $h_{\ell }(-d)$ denote the $\ell$ -part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$ . Nontrivial pointwise upper bounds are known only for $\ell =3$ ; nontrivial upper bounds for averages of $h_{\ell }(-d)$ have previously been known only for $\ell =3,5$ . In this paper we prove nontrivial upper bounds for the average of $h_{\ell }(-d)$ for all primes $\ell \geqslant 7$ , as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geqslant 3$ .

Schmidt [‘Integer points on curves of genus 1’, Compos. Math. 81 (1992), 33–59] conjectured that the number of integer points on the elliptic curve defined by the equation $y^{2}=x^{3}+ax^{2}+bx+c$ , with $a,b,c\in \mathbb{Z}$ , is $O_{\unicode[STIX]{x1D716}}(\max \{1,|a|,|b|,|c|\}^{\unicode[STIX]{x1D716}})$ for any $\unicode[STIX]{x1D716}>0$ . On the other hand, Duke [‘Bounds for arithmetic multiplicities’, Proc. Int. Congress Mathematicians, Vol. II (1998), 163–172] conjectured that the number of algebraic number fields of given degree and discriminant $D$ is $O_{\unicode[STIX]{x1D716}}(|D|^{\unicode[STIX]{x1D716}})$ . In this note, we prove that Duke’s conjecture for quartic number fields implies Schmidt’s conjecture. We also give a short unconditional proof of Schmidt’s conjecture for the elliptic curve $y^{2}=x^{3}+ax$ .

In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$ , $\gcd (m,n)=1$ and $2\nmid m+n$ , the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x,y,z)=(2,2,2)$ . In this paper, we prove the conjecture if $4\nmid mn$ and $y\geq 2$ .

Let $K$ be an algebraic number field of degree $d\geqslant 3$ , $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$ , $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$ , $a_{0}\in \mathbb{Z}$ , $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$

$\unicode[STIX]{x1D710}$

$K$

$a\in \mathbb{Z}$

$F_{0}(X,Y)\in \mathbb{Z}[X,Y]$

$\unicode[STIX]{x1D710}^{a}$

$a\in \mathbb{Z}$

$\unicode[STIX]{x1D710}$

$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$

$m>0$

$(x,y,a)\in \mathbb{Z}^{3}$

$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$

$xy\not =0$

$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$

$m$

$K$

$F_{0}$

$\unicode[STIX]{x1D710}$

$d$

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$

$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$

$I_{s,k,r}(X)$

$1\leqslant r\leqslant k-1$

$s,k\in \mathbb{N}$

$k\geqslant 3$

$1\leqslant s\leqslant (k^{2}-1)/2$

$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$

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.

Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$ , where $m$ and $n$ are positive integers with $m>n$ , $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$ . In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$ . We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$ . We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$ . Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$ .

We consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in $n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over $\mathbb{R}$ . In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$ , and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$ -adic units in $\mathbb{Z}_{p}$ . It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime  $p$ .

We improve the known upper bound for the number of Diophantine $D(4)$ -quintuples by using the most recent methods that were developed in the $D(1)$ case. More precisely, we prove that there are at most $6.8587\times 10^{29}$ $D(4)$ -quintuples.

Diophantine problems involving recurrence sequences have a long history. We consider the equation $B_{m}B_{m+d}\cdots B_{m+(k-1)d}=y^{\ell }$ in positive integers $m,d,k,y$ with $\gcd (m,d)=1$ and $k\geq 2$ , where $\ell \geq 2$ is a fixed integer and $B=(B_{n})_{n=1}^{\infty }$ is an elliptic divisibility sequence, an important class of nonlinear recurrences. We prove that the equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$ th powers in $B$ is given. We illustrate our method by an example.

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

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

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

$\ell$

$(x,y,\ell )$

$y=0$

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