The World’s Shortest Mathematics Papers
Fermat’s Last Theorem and Euler’s Conjecture
This is one of the world’s shortest mathematics papers, dating back to 1966. Though it is really short, it packs quite a lot! Let’s start with Fermat’s Last Theorem, which was discovered by Fermat and later proved by Dr. Andrew Wiles in 1993. It states, that any equation of the form:
does not have an integer solutions of x, y and z for n > 2. In the 1630s, Pierre de Fermat set a thorny challenge for mathematics with a note scribbled in the margin of a page, claiming he had a truly marvellous proof to the problem but the margin was too narrow to contain the whole of it, as he said, “Hanc marginis exiguitas non caperet”. The world was unaware of this little beast for many years following Fermat’s death, as the book, ‘Arithmeticorvm by Diaphanti,’ the problem was written on lay collecting dust. However, a few years after his death, Fermat’s son, Samuel Clément upon rediscovering this, published a new version of the book, ‘Arithmeticorvm by Diaphanti,’ containing all of Fermat’s notes (there were multitudes!) as open mathematical challenges for people to solve.
More than 350 years later, mathematician Andrew Wiles finally closed the book on Fermat’s Last Theorem, with his 109 page paper on “Modular elliptical curves and Fermat’s Last Theorem.” While 109 pages was clearly not something that could be briefed in a margin of a book, no one knows if Fermat actually knew the proof to it. In fact, people were able to find proofs to every other problem presented in the book published by Clément except for this that took 350 years. Though Fermat’s theorem remains as one of the most famous theorems of the century, there’s a similar theorem that was put forth by Euler (1707–83) who claimed, any equations of the form:
has no integer solution for k > n. While you may think that it looks very similar to Fermat’s Last theorem, it is because it actually is a generalisation of Fermat’s last theorem, however, just like Fermat’s Last theorem, this was published without any proof, which goes without saying. However, it is interesting to note that the generalisation of Fermat’s Last theorem does not actually hold true. In the paper that now holds a record as one of the shortest mathematics papers in history, Lander and Parkin proved just the same. It was observed that there was at least one solution to Euler’s conjecture, for n = 4 and k = 5, as shown in the paper:
Which was against what Euler hoped to achieve by the generalisation. Thus, this example completely breaks Euler’s conjecture and hence also holds the title, “Counterexamples to Euler’s conjecture on sums of like powers.”
This paper held the record for the shortest mathematics paper in history for some years, however the record has now been broken by a paper from Princeton University, which I will discuss in the next part of the series!
Homer says, “Thank you for reading”
