abc conjecture

41 years old · traced to 1985

For a positive integer $n$ , the radical $\operatorname{rad}(n)$ is the product of the distinct primes dividing $n$ :

\operatorname{rad}(n)=\prod_{p\mid n}p,\qquad \operatorname{rad}(1)=1.

An abc triple is a triple $(a,b,c)$ of positive integers with $\gcd(a,b)=1$ and

a+b=c.

Then $a$ , $b$ , $c$ are pairwise coprime.

Conjecture (Masser–Oesterlé). For every real $\varepsilon>0$ there is a constant $K_\varepsilon>0$ such that every abc triple satisfies

c< K_\varepsilon\,\operatorname{rad}(abc)^{1+\varepsilon}.

Equivalently, for every $\varepsilon>0$ only finitely many abc triples satisfy $c>\operatorname{rad}(abc)^{1+\varepsilon}$ . The exponent $1+\varepsilon$ cannot be replaced by $1$ : there are infinitely many abc triples with $c>\operatorname{rad}(abc)$ , for example $(1,\,3^{2^k}-1,\,3^{2^k})$ , since $2^{k+2}$ divides $3^{2^k}-1$ . A solution must prove the conjecture for every $\varepsilon>0$ . The constant $K_\varepsilon$ need not be effective.

Background

Masser and Oesterlé formulated the conjecture in 1985. Oesterlé's Bourbaki exposé gives an early account [1]. It was motivated by the Mason–Stothers theorem for polynomials [2]: if $a,b,c\in\mathbb C[t]$ are coprime, not all constant, and $a+b=c$ , then $\max\deg(a,b,c)$ is less than the number of distinct roots of $abc$ . The abc conjecture is the analogue for integers, with $\log c$ in place of degree and $\log\operatorname{rad}(abc)$ in place of the number of roots.

The conjecture says that an additive relation $a+b=c$ cannot hold among numbers built from few primes with high multiplicity. Many Diophantine statements follow from it. It implies Fermat's Last Theorem for all sufficiently large exponents and is equivalent to a modified form of Szpiro's conjecture on discriminants of elliptic curves [1]. Elkies showed that the abc conjecture over number fields implies the Mordell conjecture, with effective height bounds if the abc constant is effective [3]. Waldschmidt's survey collects many further consequences [10].

Equivalent formulations

Let $x$ be a positive integer with prime factorization
$x=p_1^{a_1}p_2^{a_2}\cdots p_n^{a_n}.$
Its radical is
$\operatorname{rad}(x)=p_1p_2\cdots p_n.$
The abc conjecture. For every $\varepsilon>0$ , there are finitely many triples of positive integers $a,b,c$ such that
$\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1,\qquad a+b=c,$
and
$\operatorname{rad}(abc)^{1+\varepsilon}<c.$
The conjecture is a central open problem in number theory and has many consequences, including conditional results about Fermat-type equations and the distribution of powerful numbers.
source: David Cushing and James Eldred Pascoe, “Powerful numbers and the ABC-conjecture”, arXiv:1611.01192 (2016).
Let $\operatorname{rad}(n)=\prod_{p\mid n}p$ be the product of the distinct primes dividing $n$ . For positive integers $a,b,c$ with $a+b=c$ and $\gcd(a,b)=1$ , the radical of $abc$ controls $c$ as follows.

The abc conjecture. For every $\epsilon>0$ , there exists a constant $C_\epsilon$ such that, for all $a,b,c\in\mathbb{Z}^+$ satisfying $a+b=c$ and $\gcd(a,b)=1$ ,
$c\leq C_\epsilon\bigl(\operatorname{rad}(abc)\bigr)^{1+\epsilon}.$
This is the abc conjecture of Masser and Oesterlé. It is used in the paper to prove that only finitely many solutions of the generalized Nagell–Ljunggren equation have inadmissible parameter triples; the conjecture itself remains open.
source: Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit and Jeffrey Shallit, “The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations”, arXiv:1707.03894 (2017).
For a positive integer $n$ , let $R(n):=\prod_{p\mid n}p$ be its radical. The abc-conjecture. For every $\epsilon>0$ , there exists a constant $C_\epsilon$ such that for all triples $(a,b,c)$ of coprime positive integers satisfying $a+b=c$ , one has
$c<C_\epsilon R(abc)^{1+\epsilon}.$
This is the classical abc-conjecture, a central open problem relating additive relations among coprime integers to the product of their distinct prime divisors.
source: Tsz Ho Chan, “Gaps between divisible terms in a^2 (a^2 + 1)”, arXiv:1906.11128 (2019).
For a nonzero integer $m$ , define its algebraic radical by
$N(m):=\prod_{p\mid m}p.$
The abc-conjecture. For every $\varepsilon>0$ , there exists a constant $C_{\varepsilon}$ such that for all nonzero coprime integers $a,b,c$ satisfying $a+b=c$ ,
$\max\{|a|,|b|,|c|\}<C_{\varepsilon}N(abc)^{1+\varepsilon}.$
This conjectural bound is invoked in the paper to control the low range of exponents $a\in[2,10^{16}]$ in estimates for prime-power values of the Ramanujan $\tau$ -function; the corresponding large-exponent range is handled without the abc-conjecture.
source: Florian Luca, Sibusiso Mabaso and Pantelimon Stanica, “On the prime factors of the iterates of the Ramanujan τ–function”, arXiv:2005.11231 (2020).
Let $\operatorname{rad}(n)=\prod_{p\mid n}p$ be the algebraic radical of $n$ . For all $\varepsilon>0$ , there exists a constant $K_{\varepsilon}$ such that whenever $A,B,C$ are coprime nonzero integers with $A+B=C$ , the ABC conjecture.
$\max\{|A|,|B|,|C|\}\le K_{\varepsilon}\operatorname{rad}(ABC)^{1+\varepsilon}.$
The ABC conjecture is a central open problem concerning the relationship between the size of coprime additive relations and the prime divisors of their product. In this paper it is recalled as background for the study of smooth divisors of terms of linearly recurrent sequences.
source: Nikhil Balaji and Florian Luca, “Terms of Lucas sequences having a large smooth divisor”, arXiv:2203.12364 (2022).
For nonzero integers $a,b,c$ , write
$\operatorname{rad}(n)=\prod_{p\mid n}p$
for the radical of $n$ . The integers $a,b,c$ are required to satisfy
$\operatorname{gcd}(a,b,c)=1,\qquad a+b=c.$
ABC conjecture. For each $\epsilon>0$ , there exists $M_{\epsilon}>0$ such that
$\max\{|a|,|b|,|c|\}\leq M_{\epsilon}\operatorname{rad}(abc)^{1+\epsilon}.$
This is a central conjecture in Diophantine geometry and number theory. In the source it is used to deduce finiteness of certain near-perfect values of polynomial sequences; its resolution is not established here.
source: Elchin Hasanalizade, “On near-perfect numbers of special forms”, arXiv:2206.10353 (2022).
For an integer $m$ , define its squarefree kernel, or radical, by
$\kappa(m):=\prod_{p\mid m}p.$
Let $a$ , $b$ , and $c$ be integers satisfying $a+b=c$ and $\operatorname{gcd}(a,b)=1$ . The abc-conjecture. For every $\epsilon>0$ , there exists a constant $C_\epsilon>0$ such that
$\max\{|a|,|b|,|c|\}\leq C_\epsilon\kappa(abc)^{1+\epsilon}.$
The conjecture is a central unproved statement in Diophantine number theory. In this paper it is noted that the conjecture on three consecutive powerful numbers follows from it.
source: Tsz Ho Chan, “Arithmetic progressions among powerful numbers”, arXiv:2210.00281 (2022).
Let $psilon>0$ . For a triple $(a,b,c)$ of co-prime positive integers satisfying $a+b=c$ , let $d$ be the product of the distinct prime factors of $abc$ . The abc conjecture. For every $psilon>0$ , there are only finitely many triples $(a,b,c)$ of co-prime positive integers where $a+b=c$ , such that $c>d^{1+psilon}$ . The conjecture is a central Diophantine statement and is the conjecture that provoked inter-universal Teichmüller theory. Its resolution remains disputed in the source's discussion, so no stronger status is assigned here.
source: Andrew Aberdein, “Deep Disagreement in Mathematics”, arXiv:2210.16488 (2022).
For a non-zero integer $a$ , let $N(a)$ be its algebraic radical:
$N(a)=\prod_{p\mid a}p.$
For any $\epsilon>0$ , there is a constant $K(\epsilon)$ depending only on $\epsilon$ such that whenever $A$ , $B$ , and $C$ are three coprime and non-zero integers satisfying $A+B=C$ , the ABC conjecture.
$\max\{|A|,|B|,|C|\}<K(\epsilon)N(ABC)^{1+\epsilon}.$
The conjecture gives a strong bound on solutions to additive equations in terms of the primes dividing their product. The paper invokes it conditionally in the proofs of Theorems 1.9 and 1.10; no resolution is supplied here.
source: Saša Novaković, “On the diophantine equation An!+Bm!=f(x,y)”, arXiv:2309.15007 (2023).
For a non-zero integer $a$ , let $N(a)$ be the algebraic radical
$N(a)=\prod_{p\mid a}p.$
For any $\varepsilon>0$ , ABC-conjecture. There is a constant $K(\varepsilon)$ depending only on $\varepsilon$ such that whenever $A$ , $B$ , and $C$ are three coprime and non-zero integers with $A+B=C$ , one has
$\max\{|A|,|B|,|C|\}<K(\varepsilon)N(ABC)^{1+\varepsilon}.$
This is a central conjecture in Diophantine number theory, relating the sizes of coprime solutions to $A+B=C$ to the algebraic radical of their product. The paper recalls it as background for finiteness results concerning Diophantine equations involving double factorials; its status is unresolved.
source: Saša Novaković, “Diophantine equations involving double factorials”, arXiv:2510.24312 (2025).
For a non-zero integer $a$ , let $N(a)$ be its algebraic radical:
$N(a)=\prod_{p\mid a}p.$
ABC conjecture. For every $\varepsilon>0$ , there is a constant $K(\varepsilon)$ depending only on $\varepsilon$ such that, whenever $A$ , $B$ , and $C$ are pairwise coprime non-zero integers satisfying $A+B=C$ ,
$\max\{|A|,|B|,|C|}<K(\varepsilon)N(ABC)^{1+\varepsilon}.$
This conjecture is used in the paper to obtain finiteness of integer solutions to equations of the form $P(x)=\prod_{i=1}^r H_{n_i}$ when $P$ is not necessarily irreducible. Its status is unresolved.
source: Saša Novaković, “The Diophantine equation P(x)=ri=1H_n_i”, arXiv:2601.16757 (2026).
For every $\varepsilon>0$ , there exists a constant $C_\varepsilon>0$ such that for all coprime nonzero integers $a$ , $b$ , and $c$ satisfying $a+b=c$ one has
$|c|\leq C_\varepsilon\,\operatorname{rad}(abc)^{1+\varepsilon},$
where $\operatorname{rad}(abc):=\prod_{p\mid abc}p$ is the product of the distinct primes dividing $abc$ . $abc$ Conjecture. The inequality above holds for every $\varepsilon>0$ with a corresponding constant $C_\varepsilon>0$ . The conjecture is used as a conditional hypothesis in the paper to prove an upper bound for the counting function of prime values of Ramanujan's tau function. Its general resolution is not established in the source.
source: David Kurniadi Angdinata, Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Simon Mahns, Aram H. Markosyan, Rithikesh Muddana, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin and Jujian Zhang, “ABC implies that Ramanujan's tau function misses almost all primes”, arXiv:2603.29970 (2026).
For each integer $n\ge 1$ , define the radical of $n$ by
$\gamma(n):=\prod_{p\mid n}p.$
The abc conjecture. For every $\epsilon>0$ , there exists a constant $C(\epsilon)$ such that for all $(a,b,c)\in\mathbb{N}^3$ with $\gcd(a,b,c)=1$ and $a+b=c$ ,
$c<C(\epsilon)\gamma(abc)^{1+\epsilon}.$
Also called the Oesterlé–Masser conjecture, this is a central conjecture in number theory and is motivated in part by the Mason–Stothers theorem. It remains open; the source uses it as the classical conjecture underlying the paper.
source: Patrick Letendre, “The abc Conjecture Revisited”, arXiv:2607.07641 (2026).
An ABC-solution is a triple $\mathbf{s}=(a,b,c)$ of distinct relatively prime integers satisfying $a+b+c=0$ , with $a$ and $b$ negative. For $n>0$ , define $\operatorname{rad}(n)$ as the product of all primes dividing $n$ , and set
$f(\mathbf{s},\epsilon)=\log(c)-(1+\epsilon)\log \operatorname{rad}(abc).$
ABC conjecture. For each $\epsilon>0$ , there exists a constant $C_\epsilon$ such that
$f(\mathbf{s},\epsilon)<C_\epsilon$
for all ABC-solutions $\mathbf{s}$ . This is the conjecture of Masser and Oesterlé, closely related to several questions in arithmetic geometry, including the arithmetic of elliptic curves. Its status is not resolved in the supplied source.
source: Jordan S. Ellenberg, “Congruence ABC implies ABC”, arXiv:math/9909098 (1999).
Let $\epsilon>0$ . For relatively prime integers $a,b,c\in\mathbb{Z}$ satisfying $a+b=c$ , define the radical of $abc$ by
$\prod_{\text{prime }p\mid abc}p.$
ABC conjecture. There exists a positive constant $C$ , depending only on $\epsilon$ , such that
$\max\{|a|,|b|,|c|\}\leq C\left(\prod_{\text{prime }p\mid abc}p\right)^{1+\epsilon}.$
This is the standard $abc$ conjecture, a central open problem in Diophantine number theory. In the paper it is used as an assumption to obtain a stronger upper bound for the relevant counting function.
source: Zafer Selcuk Aygin and Khoa D. Nguyen, “Monogenic pure cubics”, arXiv:2009.02442 (2020).
Let $a$ , $b$ , and $c$ be coprime positive integers satisfying
$a+b=c.$
For a positive integer $n$ , define its radical by
$\operatorname{rad}(n)=\prod_{p\mid n}p,$
where the product is over the distinct prime divisors of $n$ . The source also uses $q(a,b,c)$ , but does not define it in the supplied text.

The abc conjecture. For every positive real number $\varepsilon$ , the following equivalent formulations hold: there are only finitely many triples $(a,b,c)$ such that
$c>\operatorname{rad}(abc)^{1+\varepsilon},$
there is a constant $K_\varepsilon$ such that
$c<K_\varepsilon\operatorname{rad}(abc)^{1+\varepsilon},$
for all such triples, and there are only finitely many triples satisfying
$q(a,b,c)>1+\varepsilon.$
The conjecture is a central open problem in Diophantine number theory, proposed by Oesterlé and Masser and closely related to the modified Szpiro conjecture. It has numerous consequences for other conjectures and theorems, but remains unproved according to the supplied source.
source: Geoffrey B. Campbell, “Vector Partition Identities for 2D, 3D and nD Lattices”, arXiv:2302.01091 (2023).
Let $a$ , $b$ , and $c$ be positive, coprime integers satisfying
$a+b=c.$
The abc-conjecture. For every $\epsilon>0$ , there exists a constant $\kappa(\epsilon)>0$ such that
$c<\kappa(\epsilon)\operatorname{Rad}(abc)^{1+\epsilon}.$
This conjecture is the input used for the paper's bounds on arithmetic progressions of powerful numbers; the supplied text does not state whether it is resolved.
source: Prajeet Bajpai, Michael A. Bennett and Tsz Ho Chan, “Arithmetic Progressions in Squarefull Numbers”, arXiv:2302.03113 (2023).
Let $\epsilon>0$ be a fixed real number. For coprime positive integers $a,b,c$ satisfying $a+b=c$ with $a<b$ , let $\operatorname{Rad}(\gamma)$ denote the product of the distinct prime factors of the positive integer $\gamma$ .

The abc conjecture. The set of triples $(a,b,c)$ such that
$\operatorname{Rad}(abc)<c^{\frac{1}{1+\epsilon}}$
is finite.

This is a central conjecture in Diophantine analysis, asserting finiteness when the radical of $abc$ is sufficiently small relative to $c$ . The paper applies it conditionally to obtain finiteness results for digital anomalies with fixed exponent $k$ ; its resolution status is not specified in the source.
source: Samer Seraj, “Diophantine Analysis of a Digital Anomaly”, arXiv:2512.06056 (2025).

References

J. Oesterlé, "Nouvelles approches du « théorème » de Fermat", Séminaire Bourbaki, exp. 694, Astérisque 161–162 (1988), 165–186.
W. W. Stothers, "Polynomial identities and Hauptmoduln", Quart. J. Math. Oxford Ser. (2) 32 (1981), 349–370. doi:10.1093/qmath/32.3.349
N. D. Elkies, "ABC implies Mordell", Int. Math. Res. Not. 1991, no. 7, 99–109. doi:10.1155/S1073792891000144
C. L. Stewart and K. Yu, "On the abc conjecture, II", Duke Math. J. 108 (2001), 169–181. doi:10.1215/S0012-7094-01-10815-6
H. Pasten, "The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$ ", Invent. Math. 236 (2024), 373–385. doi:10.1007/s00222-024-01244-6
C. L. Stewart and R. Tijdeman, "On the Oesterlé–Masser conjecture", Monatsh. Math. 102 (1986), 251–257. doi:10.1007/BF01294603
J. D. Lichtman, "The abc conjecture is true almost always", Amer. Math. Monthly (2026). arXiv:2505.13991
S. Mochizuki, "Inter-universal Teichmüller theory IV: log-volume computations and set-theoretic foundations", Publ. Res. Inst. Math. Sci. 57 (2021), 627–723. doi:10.4171/PRIMS/57-1-4
P. Scholze and J. Stix, "Why abc is still a conjecture", manuscript (2018). PDF
M. Waldschmidt, "Lecture on the abc conjecture and some of its consequences", in Mathematics in the 21st Century, Springer Proc. Math. Stat. (2014), 211–230. doi:10.1007/978-3-0348-0859-0_13
Wikipedia, abc conjecture.

Source: Wikipedia

unsolved-problem

Posted 2 months ago · edited 5 days ago

Progress summary

Refreshed 2 months ago

Open · progress reported

The conjecture remains unproved: a claimed solution is disputed, while newer work only establishes restricted estimates and bounds on exceptional cases.

The conjecture, formulated in the 1980s, asserts that only finitely many coprime positive-integer solutions to $a+b=c$ violate the radical bound for any exponent above $1$ . Shinichi Mochizuki announced a proof in 2012, but it has not gained broad acceptance.

Known results

de Bruijn, 1962: at most $O(N^{2/3})$ exceptional triples in a cube of side length $N$ .
van Frankenhuysen: extreme examples with $c>\operatorname{rad}(abc)\exp\!\left(6.068\frac{\sqrt{\log c}}{\log\log c}\right)$ .
Browning, Lichtman, and Teräväinen, 2024: $N_\lambda(X)=O(X^{33/50})$ for fixed $\lambda\in(0,1.001)$ , a density result rather than finiteness.

Fresh attempts to settle Mochizuki’s proof have made no headway.

Current status (as of August 2026): Mochizuki’s claimed proof remains unverified and disputed, no counterexample is known, and the conjecture itself remains open despite restricted partial results.

Sources

No solutions have been posted yet.