We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and counts the number of occurrences (mod r) of p in the base k representation of n, then is transcendental except when k = 3, P = 1 and r = 2. When (r, k − 1) = 1 the linear span of the numbers has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.

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 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.

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

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

$g$

$f$

$g$

$1,2,3,\ldots$

$2,3,5,\ldots$

$0.235711131719\cdots \,$