We exhibit a sequence (un) which is not uniformly distributed modulo one even though for each fixed integer k ≥ 2 the sequence (kun) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (un) which is u.d. (mod 1) even though no subsequence of the form (ukn + j) is u.d. (mod 1) for any k ≥ 2. We prove that, if the subsequences (ukn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (un) is u.d. (mod I) if the sum of the reciprocals of the primes in P diverges. We show that this result is the best possible of its type.

Let S be the surface of the unit sphere in three-dimensional euclidean space, and let WN=(x1x2, xN)be an N-tuple of points on S. We consider the product of mutual distances and, for the variable point x on S, the product of distance from x to the points of ωN. We obtain essentially best possible bounds for maxωN p(ΩN) and for minωN maxx∈sp(x, ωN).

We say that a positive integer d has property (A) if for all positive integers m there is an integer x, depending on m, such that, setting n = m + d, x lies between m and n and x is co-prime to mn. We show that infinitely many even d and infinitely many odd d have property (A) and that infinitely many even d do not have property (A). We conjecture and provide supporting evidence that all odd d have property (A).

Following A. R. Woods [3] we then describe conditions (Au) (for each u) asserting, for a given d, the existence of a chain of at most u + 2 integers, each co-prime to its neighbours, which start with m and increase, finishing at n = m + d. Property (A) is equivalent to condition (A1), and it is easily shown that property (Ai) implies property (Ai+1). Woods showed that for some u all d have property (Au), and we conjecture and provide supporting evidence that the least such u is 2.

This paper is concerned with non-trivial solvability in p–adic integers, for relatively large primes p, of a pair of additive equations of degree k > 1: where the coefficients a1,…, an, b1,…, bn are rational integers.

Our first theorem shows that the above equations have a non-trivial solution in p–adic integers if n > 4k and p > k6. The condition on n is best possible.

The later part of the paper obtains further information for the particular case k = 5. specifically we show that when k = 5 the above equations have a non-trivial solution in p–adic integers (a) for all p > 3061 if n ≥ 21; (b) for all p execpt p = 5, 11 if n ≥ 26.

In this article we establish an estimate for a sum over primes that is the analogue of an estimate for a sum over consecutive integers which has proved to be very useful in applications of exponential sums to problems in number theory.

This paper studies quintic residuacity of primes p of the form for which the expression for 4f modulo p given in the first volume of this journal becomes indeterminate, and replaces it by a much simpler expression.

Let L be a linear differential operator with rational coefficients such that 0 is not an irregular singularity of L and that for sufficiently many p's the equation Lv = 0 has no zero solution mod p. We show that if u is a formal power series whose coefficients are p-adic integers for almost all p and if Lu is rational, then u too is rational.

We employ the Dyson's Lemma of Esnault and Viehweg to obtain a new and sharp formulation of Roth's Theorem on the approximation of algebraic numbers by algebraic numbers and apply our arguments to yield a refinement of the Davenport-Roth result on the number of exceptions to Roth's inequality and a sharpening of the Cugiani-Mahler theorem. We improve on the order of magnitude of the results rather than just on the constants involved.

Let R, S be a partition of 2, 3,… so that rational powers fall in the same class. Let (λn) be any real sequence; we show that there exists a set N, of dimension 1, so that (x + λn) (n = 1,2, …) are normal to every base from R and to no base from S, for every x ∈ N.

It is shown that any odd perfect number has a component greater than 1020.

Let $g\geqslant 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\unicode[STIX]{x1D711}$ denote Euler’s totient function, let $\unicode[STIX]{x1D70E}$ be the sum-of-divisors function, and let $\unicode[STIX]{x1D706}$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\unicode[STIX]{x1D711}$, $\unicode[STIX]{x1D70E}$, or $\unicode[STIX]{x1D706}$, then the number

$$\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray}$$

$g$

$f$

$g$

$1,2,3,\ldots$

$2,3,5,\ldots$

$0.235711131719\cdots \,$