abc conjecture
E530309
The abc conjecture is a deep and influential unsolved problem in number theory that predicts a surprising relationship between the prime factors of three integers a, b, and c satisfying a + b = c, with far-reaching consequences for many Diophantine equations.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| alsoKnownAs | Oesterlé–Masser conjecture NERFINISHED ⓘ |
| concerns |
interaction between addition and multiplication of integers
ⓘ
size of c relative to radical of abc ⓘ |
| coreIdea |
c is rarely much larger than the product of distinct prime factors of abc
ⓘ
powerful restrictions on quality of abc triples ⓘ |
| field | number theory ⓘ |
| formulatedBy |
David Masser
NERFINISHED
ⓘ
Joseph Oesterlé NERFINISHED ⓘ |
| hasConsequence |
bounds for the number of solutions to polynomial equations in integers
ⓘ
bounds on exponents in Fermat-type equations ⓘ results on Catalan-type equations ⓘ results on Pillai-type equations ⓘ results on Thue equations ⓘ results on elliptic curves over Q ⓘ results on integral points on curves ⓘ |
| hasParameter | epsilon > 0 ⓘ |
| implies |
Faltings theorem for many special cases
ⓘ
Mordell conjecture over the rationals ⓘ effective versions of Siegel’s theorem on integral points ⓘ finiteness of perfect powers in arithmetic progressions under conditions ⓘ finiteness of solutions to many Diophantine equations ⓘ results on the distribution of powerful numbers ⓘ results on the distribution of squarefree values of polynomials ⓘ strong results on integer solutions to polynomial equations ⓘ |
| importance |
central open problem in arithmetic geometry
ⓘ
far-reaching consequences in Diophantine number theory ⓘ |
| involves |
Diophantine equations
NERFINISHED
ⓘ
coprime integers a, b, c ⓘ integer exponents ⓘ prime factors of a, b, c ⓘ product of distinct prime factors ⓘ radical of an integer ⓘ |
| involvesEquation | a + b = c ⓘ |
| namedAfter |
David Masser
NERFINISHED
ⓘ
Joseph Oesterlé NERFINISHED ⓘ |
| openProblemAsOf | 2024 ⓘ |
| predicts | only finitely many triples with c > rad(abc)^{1+ε} for fixed ε ⓘ |
| relatedTo |
Fermat’s Last Theorem
NERFINISHED
ⓘ
Mason–Stothers theorem NERFINISHED ⓘ Roth’s theorem NERFINISHED ⓘ Szpiro conjecture NERFINISHED ⓘ Vojta’s conjectures NERFINISHED ⓘ |
| status | unproven ⓘ |
| subfield | Diophantine analysis NERFINISHED ⓘ |
| yearProposed | 1985 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.