In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

Let T be an algebraic torus over ℚ such that T(ℝ) is compact. Assuming the generalized Riemann hypothesis, we give a lower bound for the size of the class group of T modulo its n-torsion in terms of a small power of the discriminant of the splitting field of T. As a corollary, we obtain an upper bound on the n-torsion in that class group. This generalizes known results on the structure of class groups of complex multiplication fields.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

We develop Weyl differencing and Hua-type lemmata for a class of multidimensional exponential sums. We then apply our estimates to bound the number of variables required to establish an asymptotic formula for the number of solutions of a system of diophantine equations arising from the study of linear spaces on hypersurfaces. For small values of the degree and dimension, our results are superior to those stemming from the author’s earlier work on Vinogradov’s mean value theorem.

In this paper, we consider the simultaneous representation of pairs of positive integers. We show that every pair of large positive even integers can be represented in the form of a pair of linear equations in four prime variables and k powers of two. Here, k=63 in general and k=31 under the generalised Riemann hypothesis.

We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓa(p)=r}≪εrε for any ε>0, where ℓa(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

We study various Dirichlet series of the form ∑n≥1f(πnα)/ns, where α is an irrational number and f(x) is a trigonometric function like cot(x), 1/sin(x) or 1/sin2(x). The convergence is slow and strongly depends on the Diophantine properties of α. We provide necessary and sufficient convergence conditions using the continued fraction of α. We also show that any one of our series is equal to a related series, which converges much faster, defined in term of iterations of the continued fraction operator α↦{1/α}.

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

Let p be an odd prime. In this paper, we consider the equation and we describe all its solutions. Moreover, we prove that this equation has no solution (x,y,m,n) when n>3 is an odd prime and y is not the sum of two consecutive squares. This extends the work of Tengely [On the diophantine equation x2+q2m=2yp, Acta Arith.127(1) (2007), 71–86].

Let q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.

Let be a commutative algebraic group defined over a number field K. For a prime ℘ in K where has good reduction, let N℘,n be the number of n-torsion points of the reduction of modulo ℘ where n is a positive integer. When is of dimension one and n is relatively prime to a fixed finite set of primes depending on , we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

Let a,b,c be relatively prime positive integers such that a2+b2=c2. Half a century ago, Jeśmanowicz [‘Several remarks on Pythagorean numbers’, Wiadom. Mat.1 (1955/56), 196–202] conjectured that for any given positive integer n the only solution of (an)x+(bn)y=(cn)z in positive integers is (x,y,z)=(2,2,2). In this paper, we show that (8n)x+(15n)y=(17n)z has no solution in positive integers other than (x,y,z)=(2,2,2).

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

We give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+a), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums ∑ p≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where p is prime.

In this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package.