Math Conjectures

Fermat–Catalan Conjecture

OPENMajorConjectureProposed 1995 · Full conjecture

Canonical statement

There are only finitely many sextuples (a,b,c;m,n,k)∈Z>03×Z≥23(a,b,c;m,n,k)\in\mathbb Z_{>0}^3\times\mathbb Z_{\ge2}^3 such that
am+bn=ck,gcd⁡(a,b,c)=1,1m+1n+1k<1. a^m+b^n=c^k,\qquad \gcd(a,b,c)=1,\qquad \frac1m+\frac1n+\frac1k<1.
View source LaTeX
There are only finitely many sextuples \((a,b,c;m,n,k)\in\mathbb Z_{>0}^3\times\mathbb Z_{\ge2}^3\) such that
\[
  a^m+b^n=c^k,\qquad \gcd(a,b,c)=1,\qquad
  \frac1m+\frac1n+\frac1k<1.
\]
NT-DIO-011:[DarmonGranville1995][PoonenSchaeferStoll2005][DarmonMerel1997]
Number theoryAlgebraic geometry

Notes

The Fermat–Catalan conjecture interpolates between Fermat's last theorem and Catalan's equation. It asserts that am+bn=cka^m+b^n=c^k has only finitely many solutions in coprime positive integers a,b,ca,b,c with exponents m,n,k≥2m,n,k\ge2 satisfying 1m+1n+1k<1\tfrac1m+\tfrac1n+\tfrac1k<1, counted over all such exponent triples at once. The conjecture crystallized in 1995 in the work of Darmon and Granville [DarmonGranville1995].

For each fixed exponent triple in this hyperbolic range, Darmon and Granville proved finiteness of primitive solutions by relating the equation to curves of higher genus and invoking Faltings's theorem [DarmonGranville1995]. Individual triples have since been settled completely: Darmon and Merel disposed of Fermat-type variants via the winding quotient [DarmonMerel1997], and Poonen, Schaefer and Stoll determined all primitive solutions of x2+y3=z7x^2+y^3=z^7 through twists of X(7)X(7) [PoonenSchaeferStoll2005]. Only a short list of primitive solutions is known, each involving the exponent 22.

The obstacle is uniformity: the fixed-triple finiteness results do not combine to exclude infinitely many solutions spread across varying exponents. The conjecture, which would follow from the abcabc conjecture, remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.