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Fermat's Last Theorem
Or, more precisely, Fermat's conjecture (A conjecture is a theorem before it has been proved). This conjecture became one of the "holy grails" of mathematics, and was unsolved for hundreds of years.
Let's start by proving something similar, though not related. Pythagorus' Theorum, a specific case of the Cosine Rule, can be used to ascertain the hypotonuse of a right angled triangle:
The geometric proof for this is quite simple. If a smaller square is placed inside a larger square, we get the following shape (Note the pronumerals):
The formula for the TOTAL area of the shape can be obtained one of two ways.
1. By Using the large square:
2. Or by using the smaller square, plus the area of the four triangles:
If we equate these two equations, we get:
Simple, easy proof, but nothing to do with Fermat's Last Theorem.
The Theorem Itself
Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.
In easy-speak, no natural numbers will make that equation work apart from 0, 1 and 2. He then went on to say that he has proved it, but couldn't be bothered writing it as it wouldn not fit in the margin. Whether or not he did prove it will always remain a mystery, however the scale of the proof required would infer that there is a very slim chance that he got anywhere near it.
Infact, over 1000 false proofs were published between 1908 and 1912.
Why is it so hard to prove?
You can't just insert numbers and try it. A proof such as this one means that it must hold true for ALL numbers. If you found three numbers that made it work, then, fine, the conjecture is disproven. However, simply computing to prove the equation IS true cannot be done. If you tried every number up to 1000, there are still more numbers to be tried. If you got a computer to try every number up to 100 000 000 000 000 000 000 000, there are still more possibilities, and you are no closer to solving the problem.
To cut a long story short, The conjecture was not proven until the 1990's, When Andrew Wiles locked himself in a bedroom for seven years and put his head down. His work was based on the work of many who went before him, and there is not enough space here to cover all the mathematicians who contributed, however the important developments will be listed here:
Thanx to this site:
If anyone would like to add anything, or know more, email me firstname.lastname@example.org.