Why does n=2 work, but from n=3 onward it doesn’t?
There are infinitely many natural numbers that satisfy a²+b²=c², such as 3-4-5 (Pythagorean triples). But in 1637, French mathematician Pierre de Fermat wrote in the margin of a book that “when n is 3 or greater, there are no natural numbers satisfying aⁿ+bⁿ=cⁿ. I have found a truly wonderful proof, but the margin is too small to contain it.” Search for yourself to see whether they really do not exist.
What this lab does: Choose n and a search range, and the computer checks every possibility in that range to see whether there is a combination of natural numbers satisfying aⁿ+bⁿ=cⁿ. For n=2, you will find examples quickly, but from n=3 onward, none will appear no matter how far you search.
n (exponent)2
Search range (1 to N)30
💡 No one knows whether Fermat really had a proof. The problem remained unsolved for 358 years, even by the world’s leading mathematicians. Then, in 1994, British mathematician Andrew Wiles finally proved it using advanced modern mathematics, including elliptic curves and the modularity theorem — the proof papers alone run to more than 100 pages.