Concept
Equations seeking only integer solutions. Fermat's Last Theorem is one. Hilbert's 10th Problem (Matiyasevich 1970) proved them generally undecidable.
Understand it in one breath
Real solutions are easy; integer solutions are insanely hard. x²+y²=z² has infinitely many integer solutions (Pythagorean triples), yet x³+y³=z³ has none beyond the trivial (Fermat 1637, Wiles 1995). A tiny exponent change creates an entirely different world.
At a glance
Equation | Real solutions | Integer solutions | Decidable? |
|---|---|---|---|
ax + by = c | Infinitely many | Infinitely many (when gcd(a,b) divides c) | ✓ Euclidean algorithm |
x² + y² = z² (Pythagorean equation) | Infinitely many | Infinitely many — (3,4,5), (5,12,13)… | ✓ All can be parametrized |
x³ + y³ = z³ (FLT n=3) | Infinitely many | 0 (apart from trivial solutions) | ✓ Euler, 1770 |
xⁿ + yⁿ = zⁿ, n ≥ 3 | Infinitely many | 0 | ✓ Wiles, 1995 |
Integer solutions of a general polynomial equation | Always decidable | Undecidable | ✗ Hilbert’s Tenth Problem (Matiyasevich, 1970) |
Diophantine equations have been central to number theory for 2,300 years — from Euclid to Wiles. Most strikingly, a general algorithm cannot exist.
Key formula
Modern applications
Elliptic-curve cryptography, integer constraints in satisfiability problems, and integer programming.
Beyond MathVoyage
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