Can MathGPT solve the abc conjecture?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

DisputedNumber theory

abc conjecture

For coprime a + b = c, c is rarely much larger than the product of the distinct primes of abc.

Exact statement

\forall\varepsilon>0:\ \#\{(a,b,c):\ a+b=c,\ \gcd(a,b)=1,\ c>\operatorname{rad}(abc)^{1+\varepsilon}\}<\infty

For every ε > 0, only finitely many triples of coprime positive integers a + b = c satisfy c > rad(abc)^(1+ε), where rad(n) is the product of the distinct primes dividing n.

Status

Disputed. Shinichi Mochizuki's claimed proof (2012; published 2021) is not accepted by much of the community after Scholze and Stix's 2018 objection; it is widely treated as unresolved.

Posed: 1985, Joseph Oesterlé and David Masser.

In plain words

Example: 1 + 8 = 9, and rad(1·8·9) = 2·3 = 6 < 9, so c can exceed the radical; the conjecture says it can only exceed it by a power $1+\varepsilon$ finitely often. It connects addition with multiplication and would imply many other results, including an effective form of Fermat's Last Theorem for large exponents.

Key results so far

1985: Masser and Oesterlé formulate the conjecture.
2012: Mochizuki posts his inter-universal Teichmüller theory papers claiming a proof.
2018: Scholze and Stix identify what they consider a serious gap (Corollary 3.12).
2021: The papers are published in Publ. RIMS; the dispute is not resolved.

What our solver can and cannot do

No computer program or AI can solve the abc conjecture by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the abc conjecture say?: For every ε > 0, only finitely many triples of coprime positive integers a + b = c satisfy c > rad(abc)^(1+ε), where rad(n) is the product of the distinct primes dividing n.
Has the abc conjecture been solved?: Disputed. Shinichi Mochizuki's claimed proof (2012; published 2021) is not accepted by much of the community after Scholze and Stix's 2018 objection; it is widely treated as unresolved.
Can MathGPT solve the abc conjecture?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

J. Oesterlé, Séminaire Bourbaki 694 (1988).
P. Scholze and J. Stix, 'Why abc is still a conjecture' (2018).
S. Mochizuki, Publ. Res. Inst. Math. Sci. 57 (2021).

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