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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| abc conjecture canonical | 4 |
| abc conjecture (independently with Joseph Oesterlé) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570544 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: abc conjecture Context triple: [Fermat's Last Theorem, relatedProblem, abc conjecture]
-
A.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
D.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
E.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: abc conjecture Target entity description: 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.
-
A.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
D.
Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
-
E.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| alsoKnownAs |
Oesterlé–Masser conjecture
ⓘ
linked to:
Szpiro conjecture
|
| 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
ⓘ
Joseph Oesterlé ⓘ |
| 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
ⓘ
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
ⓘ
Joseph Oesterlé ⓘ |
| openProblemAsOf | 2024 ⓘ |
| predicts | only finitely many triples with c > rad(abc)^{1+ε} for fixed ε ⓘ |
| relatedTo |
Fermat’s Last Theorem
ⓘ
linked to:
Fermat's Last Theorem
Mason–Stothers theorem ⓘ Roth’s theorem ⓘ
linked to:
Roth theorem
Szpiro conjecture ⓘ Vojta’s conjectures ⓘ |
| status | unproven ⓘ |
| subfield |
Diophantine analysis
ⓘ
linked to:
Diophantine geometry
|
| yearProposed | 1985 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: abc conjecture Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: abc conjecture