Volume 53
, Issue 1

2017

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.

How do we find all points with integer coordinates on the hyperbola

$x^2-8xy+11y^2 = 1$?

One approach 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.