One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Fermat's Last Theorem — A 358-Year Margin
Fermat claimed a marvelous proof that would not fit in the margin; the note became public only after his death, and no proof survived. Wiles announced a result in 1993, repaired its gap with Richard Taylor in 1994, and published the two papers in 1995.
Learn the rule, then go farther with calculation or a small program.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesAndrew Wiles (with Richard Taylor)
Wolf Prize + Abel Prize + Wolfskehl Prize (set at 100,000 marks in 1908; ~£30,000 / 75,000 DM by 1997 due to inflation)
See the puzzle visually
Fermat’s single line — “I have a truly marvelous proof, but this margin is too narrow” — launched a 358-year procession. Partial cases (n=3, 5, 7) accumulated until the bridge to modular forms let Wiles finish the proof in 1995.
Problem statement
For n ≥ 3, no positive integers x, y, z satisfy x^n + y^n = z^n. (For n=2, Pythagorean triples like 3²+4²=5² exist infinitely.)
The story of this puzzle
Around 1637 Fermat wrote in a copy of Diophantus that he had a proof; the note became public in his son's 1670 edition, but no proof survived. Work on special cases generated new number theory. Frey's curve and Ribet's 1986 theorem linked a counterexample to non-modularity. Wiles worked privately from 1986, announced his result in 1993, and then repaired a gap with Richard Taylor in 1994. Their two 1995 papers proved far more structure than the margin itself suggested.
Try it yourself
Pythagorean triples (n=2) are infinite: (3,4,5), (5,12,13), (8,15,17). Find five more. Then ask: why does no such pattern exist for n=3? The answer is Fermat's Last Theorem — and it took 358 years to prove.
Beyond MathVoyage
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