Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $tR^{1/3}$ on a circle of radius $R$, for any given $t>0$. In particular we prove that any arc of length $ (40 + \frac{40}3\sqrt{10} )^{1/3}R^{1/3}$ on a circle of radius $R$, with $R>\sqrt{65}$, contains at most three lattice points, whereas we give an explicit infinite family of $4$-tuples of lattice points, $(\nu_{1,n},\nu_{2,n},\nu_{3,n},\nu_{4,n})$, each of which lies on an arc of length $ (40 + \frac{40}3\sqrt{10})^{\smash{1/3}}R_n^{\smash{1/3}}+o(1)$ on a circle of radius $R_n$.

Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any strictly increasing sequence of positive integers satisfying $s_n\le n^{15/14+o(1)}.$ In this paper we give a version of the large sieve inequality for the sequence $\lambda^{s_n}.$ In particular, we obtain nontrivial estimates of the associated trigonometric sums ``on average" and establish equidistribution properties of the numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$, modulo $p$ for most primes $p.$

Let $E$ be an elliptic curve defined over $\Q$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p \leq x$ for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}} {(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is $\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards a 1976 Lang--Trotter conjecture.

We investigate exceptional sets in the Waring--Goldbach problem. For example, in the cubic case, we show that all but $O(N^{79/84+\epsilon})$ integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than~3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

Let $\calS$ denote the set of integers representable as a sum of two squares. Since $\calS$ can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that $\calS$ has many properties in common with the set of prime numbers. In this paper we exhibit ``unexpected irregularities'' in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of $\calS$ than expected, and infinitely many intervals containing considerably fewer than expected.

There are infinitely many triplets of primes $p,q,r$ such that the arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$, ${q+r\over2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, ${p+q+r\over3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.}