## Can you find a simple proof for this statement?

I thought more about the last question I added into the addendum of the Numberphile, Graph theory and Mathematica post

It can be succinctly stated as:

$(\forall m\in\mathbb{Z}, m\ge 19) (\exists p,q\in\mathbb{Z}, 1\le p,q such that $\sqrt{p+m}\in\mathbb{Z}$ and $\sqrt{q+m}\in\mathbb{Z}$.

In words:

For all integers m, greater than 19, there are two other distinct positive integers less than m such that the sum of each with m, when square rooted is an integer.

What is the shortest proof you can find for this statement?

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## Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.

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## Posts in Shona, Sepedi and IsiXhosa and…

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## The Strength in Numbers: The New Science of Team Science – By Barry Bozeman & Jan Youtie, a review

NB. I was sent this book as a review copy.

If you want insights into what makes a good collaboration dream-like and a bad collaboration nightmarish, this is the book for you.

In short, The Strength in Numbers details an extremely important piece of research, with reference to many other studies, which aims to analyse collaborations within STEM, and figure out not only measures of collaboration effectiveness, but also ways to make your own collaborations more likely to succeed.

Academia is a funny old game, where there is extensive training in certain aspects of the job (the fundamental tools of science, for instance), and others are left to the researcher to try and piece together as they go along. Some obvious and frequent examples of these are:

• How to write and give talks effectively
• How to mentor young researchers
• How to best disseminate your own knowledge

and perhaps most importantly, how to create an effective collaboration.…

## Graph Theory, Numberphile and Mathematica

Edit: I made a mistake with some of the language here. A comment from a true graph theorist:

“Hamiltonian” usually means there’s a hamilton/hamiltonian cycle. Graphs with a hamilton path are “traceable”. Hamiltonian implies traceable, but not conversely.

Thus, I have edited below accordingly.

——————————-

There was a nice video up on Numberphile about a problem which could easily be explained to a school student, and yet we don’t yet know the answer to it. See the videos here:

and

The game is the following: Given a sequence of consecutive integers, draw a graph where the nodes are the integers and there is an edge between each integer if their sum is a square number.

If we take the numbers from 1 to 12, then the following would be the associated graph (note that there are three disconnected pieces of it).

Note that this is a disconnected, undirected graph. Looking at some of the edges.…

## Welcome to Mathemafrica

If you’re here from Juan Klopper’s YouTube channel, then welcome! If you’re not, then welcome!

I wanted to point new readers to a few of the different types of posts that we have here on Mathemafrica. Please, if there are some topics that you would like us to write about, let us know. We’re always happy to produce new content and especially to have new writers (if you enjoy writing about mathematics and are in Africa or have links to Africa, then let us know in a comment and we can sign you up as a contributor).

Here are some links to some of the types of posts that you will find on Mathemafrica:

## Brazil Delta Conference 2017, R. Nazim Khan: Attendance is important!

Attendance: The mismatch between academics and students. Who is right?

R. Nazim Khan, University of Western Australia

Presentation at the 11th Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics (‘Brazil Delta 2017 for short)

Students don’t attend from the 1st lecture – how do they know if the lecturer is good or not? Something else is afoot.

Lectures are recorded, students watch the recordings rather than attending.

Low attendance is correlated to high failure rate.

Reasons for low attendance:

• Maths is not their major
• Lecturing is not done well
• Students believe they can pass without attending
• When students have jobs and work comes up, they prioritise work over attendance
• Attendance is higher when there is assessment in class
• Research on attendance …

This study:

Does attendance affect performance? Who attends class? Attitudes on attending, staff and students?

Business statistics course, records of attendance, survey, marks, demographics, high school marks.…

## Brazil Delta Conference 2017, Barbara Miller-Reilly: A Tale of Two Journeys

A tale of two journeys

Barbara Miller-Reilly and Charles O’Brien

Presentation at the 11th Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics (‘Brazil Delta 2017 for short)

20 years ago Charles was a young business man with a fear of maths; Barbara was a teacher of mathematics-avoident adults and working on a PhD.

Recently Charles asked Barbara for help with maths again. They decided to co-author a paper on their journeys.

Barbara:

• Left teaching to look after her 3 children
• Did part-time teaching, e.g. course on Maths Anxiety, where she learnt many skills
• Started job-share university work
• Developed maths and statistics learning
• Researched programmes encouraging girls in STEM
• Started PhD in early 50’s

Charles

• Lower stream in maths at aged 13
• Failed all maths tests in high school
• Left his brokering job due to fear of maths
• Wondered if he has a maths version of dyslexia
• Said maths was like skiing on blue ice with no edges, blindfolded.