Fermat's Last Theorem (41 page)

5 years later a son was born,

L
⁄
₂
was the life span of the son,

4 years were spent in grief before he died.

The length of Diophantus' life is the sum of the above:

We can then simplify the equation as follows:

Diophantus died at the age of 84 years.

In order to weigh any whole number of kilograms from 1 to 40 most people will suggest that six weights are required: 1, 2, 4, 8, 16, 32 kg. In this way, all the weights can easily be achieved by placing the following combinations in one pan:

However, by placing weights in both pans, such that weights are also allowed to sit alongside the object being weighed, Bachet could complete the task with only four weights: 1, 3, 9, 27 kg. A weight placed in the same pan as the object being weighed effectively assumes a negative value. Thus, the weights can be achieved as follows:

A Pythagorean triple is a set of three whole numbers, such that one number squared added to another number squared equals the third number squared. Euclid could prove that there are an infinite number of such Pythagorean triples.

Euclid's proof begins with the observation that the difference between successive square numbers is always an odd number:

Every single one of the infinity of odd numbers can be added to a particular square number to make another square number. A fraction of these odd numbers are themselves square, but a fraction of infinity is also infinite.

Therefore there are also an infinity of odd square numbers which can be added to one square to make another square number. In other words there must be an infinite number of Pythagorean triples.

The dot conjecture states that it is impossible to draw a dot diagram such that every line has at least three dots on it.

Although this proof requires a minimal amount of mathematics, it does rely on some geometrical gymnastics, and so I would recommend careful contemplation of each step.

First consider an arbitrary pattern of dots and the lines which connect every dot to every other one. Then, for each dot, work out its distance to the closest line, excluding any lines which go through it. Thereby identify which of all the dots is closest to a line.

Below is a close-up of such a dot
D
which is closest to a line
L.
The distance between the dot and the line is shown as a dashed line and this distance is smaller than any other distance between any other line and a dot.

It is now possible to show that line
L
will always have only two dots on it and that therefore the conjecture is true, i.e. it is impossible to draw a diagram such that every line has three dots on it.

To show that line
L
must have two dots, we consider what would happen if it had a third dot. If the third dot,
D
_A
, existed outside the two dots originally shown, then the distance shown as a dotted line would be shorter than the dashed line which was supposed to be the shortest distance between a dot and a line. Therefore dot
D
_A
cannot exist.

Similarly, if the third dot,
D
_B
, exists between the two dots originally shown, then once again the distance shown as a dotted line would be shorter than the dashed line which was supposed to be the shortest distance between a dot and a line. Therefore dot
D
_B
cannot exist either.

In summary, any configuration of dots must have a minimum distance between some dot and some line, and the line in question must have only two dots. Therefore for every configuration there will always be at least this one line with only two dots – the conjecture is true.

The following is a classic demonstration of how easy it is to start off with a very simple statement and then within a few apparently straightforward and logical steps show that 2 = 1.

First, let us begin with the innocuous statement

Then multiply both sides by
a
, giving