The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus.

Let $f \in \Z[x]$ be a polynomial of degree d. The paucity of non-trivial positive integer solutions to the equation $f(x_1)+f(x_2)=f(x_{3})+f(x_{4})$ is established, provided that $d \geq 7$. Also the corresponding situation is investigated for equal sums of three like polynomials.

it is shown that, for all large $x$, there are more than $x^{0.33}$ carmichael numbers up to $x$, improving on the ground-breaking work of alford, granville and pomerance, who were the first to demonstrate that there are infinitely many such numbers. the same basic construction as that employed by these authors is used, but a slight modification enables a stronger result on primes in arithmetic progressions based on a sieve method to be employed.

This paper aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces. The final two sections of the paper outline results of the author and Kirwan on the cohomology rings of certain of the spaces described earlier (formulas for intersection numbers that were discovered by Witten (Commun. Math. Phys. 141 (1991) 153–209 and J. Geom. Phys. 9 (1992) 303–368) and given a mathematical proof by the author and Kirwan (Ann. of Math. 148 (1998) 109–196)).

Thanks to Szemerédi's theorem on sets with no long arithmetic progressions, an elementary trick is used here to show that for a given positive integer $h$ and a given set $U$ of residue classes modulo $n$ with positive density, there exists a dense subset $V$ of $U$; that is, $U\smallsetminus V$ is very small, such that the sumset $hV$ is included in the restricted sumset $h\times U$. The next step is to obtain information on the structure of $V$ from Kneser's theorem on the sum of sets in an abelian group, and to use this for studying the structure of $U$ itself. Finally, this idea is used in the paper to derive some new values of a function related to the Erdős–Ginzburg–Ziv theorem.

Let $G$ be an additively written abelian group, and let $S$ be a sequence in $G \setminus \{0\}$ with length $|S| \ge 4$. Suppose that $S$ is a product of two subsequences, say $S = B C$, such that the element $g+h$ occurs in the sequence $S$ whenever $g \cdot h$ is a subsequence of $B$ or of $C$. Then $S$ contains a proper zero-sum subsequence, apart from some well-characterized exceptional cases. This result is closely connected with restricted set addition in abelian groups. Moreover, it solves a problem on the structure of minimal zero-sum sequences, which recently occurred in the theory of non-unique factorizations.

A new lower bound on $\max \{|A+A|,|A\cdot A|\}$ is given, where $A$ is a finite set of complex numbers.

For any positive integer n we let $P(n)$ be the largest prime factor of n. We improve and generalize several results of P. Erdős and C. Stewart on $P(n!+1)$. In particular, we show that $\limsup_{n \to \infty}P(n!+1)/n \ge 2.5$, which improves their lower bound of $\limsup_{n \to \infty} P(n!+1)/n >2$.

In this paper, the theory of $\epsilon$-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant $\mu(\X,G,V)$ associated to certain symplectic motives of weight one. The relationship between this invariant and $w_2(\pi)$ (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Noguès, Erez and Taylor) is also discussed.

a proof is given to show that for an inner form of $\gl_n$ over a global field of zero characteristic, there exist only a finite number of automorphic representations with fixed local factor (up to equivalence) at almost every place. what is new in comparison to earlier work (see a. i. badulescu and p. broussous, ‘un théorème de finitude’, compositio math. 132 (2002) 177–190) is the case when the local factors are not fixed at the infinite places, as well as the statement of the result for the automorphic spectrum, rather than the cuspidal one.

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on $\SL_2(\ZZ)$. As a consequence, spaces $M_k$ are identified, in which there are universal $p$-divisibility properties for certain $p$-power coefficients. As a corollary, let $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.

Let $(a_n)_{n\ge 0}$ be a sequence of complex numbers, and, for $n\,{\ge}\,0$, let \[ b_n=\sum_{k=0}^n{n\choose k}a_k\quad\hbox{and}\quad c_n=\sum_{k=0}^n{n\choose k}(-1)^{n-k}a_k. \] A number of results are proved relating the growth of the sequences $(b_n)$ and $(c_n)$ to that of $(a_n)$. For example, given $p\,{\ge}\,0$, if $b_n\,{=}\,O(n^p)$ and $c_n\,{=}\,O(e^{\epsilon\sqrt{n}})$ for all $\epsilon\gt0$, then $a_n\,{=}\,0$ for all $n\gt p$. Also, given $0\lt\rho\lt1$, then $b_n,c_n\,{=}\,O(e^{\epsilon n^\rho})$ for all $\epsilon\gt0$ if and only if $n^{1/\rho-1}|a_n|^{1/n}\to 0$. It is further shown that, given $\beta\gt1$, if $b_n,c_n\,{=}\,O(\beta^n)$, then $a_n\,{=}\,O(\alpha^n)$, where $\alpha\,{=}\,\sqrt{\beta^2-1}$, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proofs are a Phragmén–Lindelöf theorem for entire functions of exponential type zero, and an estimate for the expected value of $e^{\phi(X)}$, where $X$ is a Poisson random variable.

In this article, the authors collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. They start with a sketch of Milne's proof of two of these conjectures, and they also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of, respectively, $4s^2$ or $4s(s+1)$ triangular numbers. This approach is inspired by Zagier's proof of the Kac–Wakimoto formulas via modular forms. The survey ends with recent conjectures of the first author and Chua.

A lower and an upper bound are proved for the maximum size of a subset of $\mathbb{Z}/p\mathbb{Z}$ containing no solutions of a fixed linear equation.

Define a sequence $(s_{n})$ of two-variable words in variables $x$, $y$ as follows: $s_{0}(x,y)=x$, $s_{n+1}(x,y)=[s_{n}(x,y)^{-y},s_{n}(x,y)]$ for $n\geq0$. It is shown that a finite group $G$ is soluble if and only if $s_{n}$ is a law of $G$ for all but finitely many values of $n$.

The existence of Salem numbers of every trace is proved; the nontrivial part of this result concerns Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, has enabled the construction to be refined so as to avoid cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. It is also shown how this new interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbitrary trace than was previously known.

let $e/\mathbb{q}$ be an elliptic curve. for a prime $p$ of good reduction, let $e(\mathbb{f}_p)$ be the set of rational points defined over the finite field $\mathbb{f}_p$. denote by $\omega(\#e(\mathbb{f}_p))$ the number of distinct prime divisors of $\#e(\mathbb{f}_p)$. for an elliptic curve with complex multiplication, the normal order of $\omega(\#e(\mathbb{f}_p))$ is shown to be $\log \log p$. the normal order of the number of distinct prime factors of the exponent of $e(\mathbb{f}_p)$ is also studied.

Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.

for a commutative ring $r$, let $\nil(r)$ be the set of all nilpotent elements of $r$, $z(r)$ the set of all zero divisors of $r$, and $t(r)$ the total quotient ring of $r$. set $\mathcal{h} = \{r \mid r$ is a commutative ring and $\nil (r)$ is a divided prime ideal of $ r \}$. for a ring $r \in \mathcal{h}$, let $\phi : t(r) \longrightarrow r_{\nil (r)}$ be such that $\phi(a/b) = a/b$ for every $a \in r$ and $b \in r\backslash z(r)$. a ring $r$ is called a zpui ring if every proper ideal of $r$ can be written as a finite product of invertible and prime ideals of $r$. this paper gives a generalization of the concept of zpui domains (which was extensively studied by olberding) to the context of rings that are in the class $\mathcal{h}$. let $r \in \mathcal{h}$. if every nonnil ideal of $r$ can be written as a finite product of invertible and prime ideals of $r$, then $r$ is called a nonnil zpui ring; also, if every nonnil ideal of $\phi(r)$ can be written as a finite product of invertible and prime ideals of $\phi(r)$, then $r$ is called a nonnil$\phi$-zpui ring. the theory of $\phi$-zpui rings is shown here to resemble that of zpui domains.

A new estimate for the exponential sum with square-free numbers is established. This result is applied to the problem of finding the number of representations of a large integer as a sum of three square-free numbers.