The Maths Theorem to have the Greatest Number of Incorrect Proofs OF ALL TIME
With n, x, y, z ∈ N (meaning: x,y,z are all positive whole numbers) and n > 2 the equation xn + yn = zn has no solutions.
Looks easy, right? Anyone who did high school maths would remember Pythagoras and think they could solve this theorem. But it’s not that simple, it had stumped mathematicians for 358 years.
The theorem is entitled “Fermat’s Last Theorem”, it is a notoriously difficult problem. It was first proposed by Fermat in the margins of his copy of ‘Arithmetica ’ in 1637, where he famously said “assuredly found an admirable proof of this, but the margin is too narrow to contain it.”
It is a more general version of the Pythagorean equation, a2 + b2 = c2 , which we all know has infinite solutions of positive integers (Pythagorean triads, fun fact this year we have a Pythagorean triad day, 15/8/17). But Fermat’s theory replaces the ‘2’ with n > 2 and says that there are no integer solutions for exponents greater than 2. Now just to prove (or disprove) for all cases.
The theorem has the greatest number of incorrect proofs due to the approachable nature of the problem, anyone who did maths in high school maths know Pythagoras and felt they could give it a go to win the prize money (and the celebrity, to solve Fermat’s last theorem is every mathematicians dream).
The centuries following its proposition many attempts were made at solving the unassuming theorem, with specific cases proven by hand and the computer (once it was invented). But for a mathematical theorem to be proven it needs a general proof, knowing that for all primes up to 4 million the theorem is true, is not sufficient in maths.
In 1993, it was proved by British mathematician Andrew Wiles (and published in 1995). The proof is over 100 pages long and spread between 2 papers. His proof draws from work by others before him. The techniques he used are much more advanced than the techniques Fermat would have had access to. This makes one think of how Fermat solved it in 1637, as Wiles proof is valid but it could not have been what Fermat had in mind. Due to this, there is still a sense of mystery around this theorem. He is credited with solving it and has received numerous awards for the proof as it is a thing of beauty.
Why have I written a blog post on this solved maths theorem you may be thinking, well I feel compelled to share the story of Fermat as not only has this problem caused headaches to mathematicians but it also was responsible for the development of whole new branches of maths. Algebraic number theory blossomed in the quest to prove Fermat’s theorem, it has also inspired generations of mathematicians like Wiles to pursue the field of mathematics. I enjoy knowing that there is beautiful maths being done because of some French lawyer writing in the margins of his maths book in the 17th century.