Fermat on arithmeticians

Thanks to Sol Robeson ( in PI ) we call mathematicians who lost it "numerologists".

In 1657, Fermat challenged William Brouncker, of Castle Lynn in Ireland, and John Wallis to find integral solutions to the equations $$x^2 − 151y^2 = 1$$ and $$x^2 − 313y^2 = −1.$$ He ( Fermat ) cautioned them not to submit rational solutions because even the lowest type of arithmetician could devise such answers.

"An Introduction to Diophantine Equations, A Problem-Based Approach, Andreescu, Andrica & Cucurezeanu, Springer 2010"

Considering that Fermat used the qualification the lowest type of arithmetician there must have been a ranking in the computational branch those days. Until at least WW2, a computer, was the job description of someone who did "computational work" in banking, insurance, trading, logistics and what have you. Jobs like that exist even now, think of the actuarial sciences, but most of them if not all require a degree in mathematics. I am not sure but I suppose that in Fermat's days there must have been people responsible for the basic addition and multiplication type of calculations. Fermat called them "arithmeticians, of the lowest kind".

I am speculating of course. Fermat could have been a terrible arrogant man looking down on the working class. Considering that he was not a mathematician himself but that he wrote, on his own initiative, letters to the great minds of his time says at least something of his self-image.

Link: My previous post on Fermat

Continued fractions (3)

Each rational number can be represented as a finite continued fraction (FCF) and each FCF represents a rational number. We have seen how to calculate the rational number from a given FCF, in this post we show how to calculate the FCF for any rational number.

For example, the FCF representation of $\frac{17}{13}$ can be calculated as follows:

$a$	$q$	$b$	$r$
$17$	$1$	$13$	$4$
$13$	$3$	$4$	$1$
$4$	$4$	$1$	$0$

The value of the FCF is contained in the second column from top to bottom: $\frac{17}{13}$ is $\left[ 1,3,4 \right]$. This is clearly an application of Euclid's algorithm for calculating the GCD of two integers. The algorithm for calculating the GCD stops at row $3$ but by adding one more row containing $4 = 4 \times 1 + 0$ the column containing the FCF is complete.

See also:
- Continued fractions (1)
- Continued fractions (2)
- Continued fractions (2a)

Contined fractions (2a)

I found a better way to present the table which shows the algorithm for calculating continued fractions:

$k$	$a_k$	$p_k$	$q_k$	$C_k$
$-1$		$0$	1
$0$		$1$	$0$
$k$	$a_k$	$a_k \cdot p_{k-1} + p_{k-2}$	$a_k \cdot q_{k-1} + q_{k-2}$	$\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also:
- Continued fractions (1)
- Continued fractions (2)

Continued fractions (2)

A finite continued fraction (FCF) is a map $$f: \mathbf{N}^m \rightarrow \mathbf{Q} $$ $$\left( a_1, a_2, \cdots a_m \right) \mapsto a_1 + \frac{1}{a_2 + \frac{1}{\ddots + \frac{1}{a_m}}}$$

Continued fractions are calculated by creating a table of convergents, as follows:

$k$	$a_k$	$p_k$	$q_k$	$C_k$
$-1$		$0$
$0$		$1$	$0$
$1$	$a_1$	$a_1 \cdot p_0 + p_{-1}$	$1$	$\frac{p_1}{q_1}$
$k$	$a_k$	$a_k \cdot p_{k-1} + p_{k-2}$	$a_k \cdot q_{k-1} + q_{k-2}$	$\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also: Continued fractions (1)

Study Tip - 2

Do you know any professional musicians, dancers perhaps? History shows that art and mathematics thrive in the same places. I, sadly, don't. Although yesterday I witnessed a pianist's daily practice routine. It started with loosening up the muscles. Then, slowly, player and instrument become one sound generating machine. To me this explained why musicians can have such deep relationships with their instruments. The musicians we see on stage ( all of them, not just the 'stars' ) have practiced at least ten-thousand hours to reach that level. I don't think it's such a bad guess to say that, any mathematician who is at the forefront and is creating new mathematics, carries at least the same weight of practice hours on his belt as a professional in the arts or an athlete.

Tip 2: Exercise daily

Work on a difficult exercise every day. Find a booklet with Olympiad level exercises or any book with exercises that are a challenge for -you-. Exercises you can do won't make you better. Hard exercises do.

Continued fractions (1)

Generating functions are "mathematical data structures" that can store an infinite amount of data.

For example $$\frac{1}{1-x} = \left\{ 1,1,1, \cdots \right\}$$ and $$\frac {1}{1-x-x^2} = \left\{ 1,1,2,3,5,8,13, \cdots \right\}$$ nicely represents the Fibonacci series. ( The existence of tools like the GF's made me sort of addicted on mathematics. ) If you think this is the most compact way to describe the Fibonacci series, then let mathematics surprise you. The most compact way to describe the Fibonacci series is $$\left[ <1> \right]$$ which means $$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots }}}.$$ Objects like this are called continued fractions, more on these and why $\left[ <1> \right]$ is related to the Fibonacci series in the next post.

Creating new mathematics ( ... )

Suppose your assignment was to teach a group of friendly aliens, just arrived from the Pleiadians, the rules of the game of chess. A student has completed the course with success if he is able to play a game according to the rules. That's not too difficult you think considering the number of six year olds able to play a descent game of chess. Your study materials are: lots of chalk and a blackboard. No chess pieces, no boards are available in class. Your teaching assistant will type your lecture as you speak, therefore it is not allowed to use drawings or symbols that can't be typed instantly.

This may seem difficult ( it is ), but compare it with the creation of new mathematics (...)

Mathematics is a parallel universe which we can enter with our mind only. Although bodiless, exterior, we are free to travel in this spectacular universe. When we come back however we lack the words to describe our observations, to communicate what we have seen with our mental ( mathematical ) eyes. Each and every observation must be recorded and analyzed before we can attempt to describe it. Definition by definition we try to create a consistent picture of what we have 'seen'. In this notion of mathematics, for example the number e always existed, it just took an Euler to describe it properly. Obviously the mathematical world is not some parallel library where we can go to and lookup the answers to the current unsolved problems. - In a BBC documentary about Andrew Wiles and Fermat's last theorem, Wiles describes his research as entering some space, then by touching things by hand in the dark he had to form a mental picture of what's in that space, and so on.

Describe what you observe (...)

Study tip - 1

( For a while I have been thinking about writing -the- great (...) post listing zillions of study tips. Since it is not hard to imagine such a post will never be written I just start with tip 1 ( in random order ), then see how far I will get and maybe, one day, compile them into one post or page. )

Tip-1: The next item on the (study-)list

If you want to start studying immediately in the time you have allocated for study make sure you know -exactly- what you are going to do when you start. Don't lose time on deciding if you are going to read, revise, do exercises, work on assignments or whatever it is that you do for studying. The best time to plan a session is at the end of each study session. This already structures a session into study / plan next session. This plan can be as short as 'Do TMA questions 2 and 3'. Or 'Read pages 12-28'. Very quickly go through it and write your plan down in your agenda ( whatever system you use ). Visualize yourself starting the next session and starting with these tasks. - The trick is that your subconscious already starts working on it. Programs, prepares you for the task. Next session, starting the task will be easy and enjoyable. It works. Make a habit of it.

Mandelbrot fractals in 3D

The laptops we use today are extremely powerful machines if you measure them against the standards of a decade ago. In these days producing ( rendering ) Mandelbrot fractals was hard work for any computer. Also, Mandelbrot fractals were 2D, by definition, case closed. Experimenting with '3d-type-of-Mandelbrots' was impossible due to the limitations of the hardware. A lot has happened since then. - Daniel White created a website about the topic, called 'The unravelling of the Real 3D Mandelbulb" where he explains the interesting ( and surprising ! ) history of 3D Mandelbrots.

Exercise

Exercise:

Find $x, y$ such that $$\frac{1}{x} + \frac{1}{y} = \frac{1}{pq}$$ where $x,y \in \mathbf{Z}$ and $p,q$ are prime.

Hint: there are nine different solutions. I'll publish the method and solution on request ( comment ).

About mathematics at the Open University

There are of course many differences between studying ( mathematics ) at the Open University and studying math at a brick university. The main difference is of course the main method of delivering knowledge: course booklets versus lectures with accompanying lecture notes. A brick university course is often based upon some textbook. Homework includes reading assignments and exercises. An Open University booklet is a mix of theory, worked examples and exercises. If the method of presentation matches the way you like to learn math following a course is easy. - The way mathematics is presented in textbooks (at the advanced undergraduate or graduate level ) is however completely different. If you learn all your math from Open University booklets this may come as a shock, since the skill to read mathematics books hasn't been developed. Compare a graduate math book in your field of interest with one of the level 3 booklets to see what I mean. Or to put it differently: you have not been initiated in the ( secret ) protocols of how mathematicians communicate.

The challenge can best be met by attempting to solve the exercises without recourse to the hints. The density of information in the text is rather high; a newcomer may need one hour for one page. Make sure to have paper and pencil at hand when reading the text.

Wolfgang Rautenberg in "A Concise Introduction to Mathematical Logic 3rd edition, Springer 2010, preface"

When nerds fall in love...

Most shapes can be described by one or more equations, the human imagination does the rest. The equation (x^2 + 9/4 y^2 + z^2 - 1)^3 - x^2 z^3 - 9/80 y^2 z^3 == 0 describes a surface representing the form of a heart.

Click to enlarge

( From an idea in the Mathematica docs.)