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If you are working towards a maths A-level, you will certainly have come across exam questions that instruct you to "show your working". This is an excellent habit to get into, as the following cautionary tale shows.
Pierre de Fermat, 1607-1655, did important work in calculus, geometry and probability, but is best remembered today for his "Last Theorem". The theorem is simple to state:
If a, b, c and n are positive whole numbers, the equation aⁿ + bⁿ = cⁿ has no solutions when n > 2.
Obviously it has solutions for n=1: you can easily choose a, b and c such that a + b = c, for example 1 + 2 = 3. And for n=2, we have 3² + 4² = 5². But try it with cubes or higher powers; it can't be done.
After Fermat died, it was discovered that he had written in the margin of a printed book that "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." But either he never wrote his demonstration down, or it was lost. Was he joking or mistaken, or had he actually discovered a proof? Nobody knew, because he had failed to show his working.
Mathematicians set to work, but it took over 300 years for a correct proof to be worked out. And it certainly would not have fitted in any margin. It consisted of two long papers, which together took up a whole issue of the Annals of Mathematics, and involved many techniques that were only developed long after Fermat's time. So it cannot have been what Fermat found.
The author of the proof, Andrew Wiles, received many awards, but missed out on the Fields Medal, the mathematical counterpart of the Nobel Prize. The Medal is only awarded to mathematicians under 40, and it took Wiles so long to prove the theorem that he was 41 by the time he finally succeeded.
I like to think Fermat would have been at least a little embarrassed at all the trouble his marginal note caused. Or was his "truly marvelous demonstration" actually correct, but so clever that no other mathematician has ever stumbled across it?
Perhaps if Andrew Wiles had had some good one-to-one lessons while he was at school, he would have got off the mark faster and proved the theorem before he turned 40. Somehow I doubt that, but maths tutoring can still be very helpful. If you are interested, please contact me. I mainly teach A level, but will also take GCSE students intending to study STEM subjects at A level.
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