Volume 53
, Issue 1


Dear Readers, welcome to this year’s first issue of Parabola, dedicated to my colleague Peter Brown. In this issue you will find three excellent articles, beautifully-set problems and, as something new, a book review.

It was quite a surprise when it was recently found that Friedman numbers have a density of 1 within the integers.
In this paper, we describe how this result is reached.

Let us begin with a question: Find all the points with integer coordinates on the hyperbola $x^2 - 8xy + 11y^2 = 1$.
How do we find all such points? One approach to this is to use continued fractions.

Our system for writing integers relies on ten symbols. When we write an integer that is less than ten, the rule is easy: write the corresponding symbol; for example, “nine” is expressed as “9”. However, for integers greater than or equal to ten, the rules are more complicated.

The book under review, Mathematical Doodlings - Curiosities, conjectures and challenges is a personal and passionate affair. Most of the book forms an ode to numbers and their patterns, a lifelong love affair that the author has enjoyed as non-professional mathematical doodler and thinker.

Q1521 Solve the equation $\sqrt{x + 20} + \sqrt{x} = 17$.

Q1511 In a certain country, every pair of towns is connected by a highway going in one direction but not by a highway going in the other direction. A town is central if it can be reached from every other town either directly, or with just one intermediate town.