Fermat's Last Theorem

What just happened?

For n=2 the equation has infinitely many solutions in natural numbers — these are ordinary Pythagorean triples. For every n greater than 2 not a single solution exists in positive integers.

Fermat's Last Theorem:

has no solutions in positive natural numbers x, y, z when the exponent n is an integer greater than 2. For n=2 this is just the Pythagorean theorem:

350 years without a proof

Pierre de Fermat wrote this theorem in the margin of a book around 1637, adding the famous note that he had “a truly marvelous proof, which this margin is too narrow to contain”. For over 350 years no mathematician was able to reconstruct it — today it is thought that Fermat was mistaken or had a proof that was wrong or incomplete.

Andrew Wiles and the proof of 1994/1995

The British mathematician Andrew Wiles worked on the proof in secret for years and presented it in 1993 — it turned out to contain a gap. After a year of corrections, with the help of Richard Taylor, in 1994 he presented a complete, correct proof (published in 1995), using the advanced theory of elliptic curves and modular forms (the Taniyama-Shimura-Weil conjecture). The proof ran to over 100 pages. For this achievement Wiles received the Abel Prize in 2016 — one of the most important equivalents of the Nobel Prize in mathematics.

Why this matters for your exam:

You don't prove Fermat's Last Theorem on an exam, but it illustrates a key idea of mathematics: a conjecture is easy to state, while a proof can be incomparably harder. Exams test the same logic on simpler examples — checking whether an equation has solutions in a given set of numbers is a typical number-theory problem.

Remember: x^n+y^n=z^n has solutions in natural numbers only for n=2 (Pythagorean triples) — for every n>2 there are none. This is Fermat's Last Theorem, proved by Andrew Wiles only after 350+ years.