Quantitative bounds for the exceptional set in the abc conjecture

For coprime positive integers with $a+b=c$ , the abc conjecture of Masser and Oesterlé asserts that only finitely many triples have radical $\operatorname{rad}(abc)$ smaller than $c^{1-\varepsilon}$ , for each fixed $\varepsilon>0$ . Equivalently, the exceptional set $\mathcal{E}(N)$ of such triples up to height $N$ should be uniformly bounded in $N$ . This endpoint is equivalent to the abc conjecture itself and is wide open; the tractable open core is quantitative: after six decades at the folklore bound $|\mathcal{E}(N)|\ll N^{2/3}$ (from de Bruijn's 1962 estimates for radicals), power-saving improvements began in 2024, with the current best exponent $3/5$ for the counting function $N_\lambda(X)$ of triples with $\operatorname{rad}(abc)<c^\lambda$ , $\lambda<1$ . The problem is to push this exponent below $3/5$ , and ultimately to decide whether $N_\lambda(X)$ is bounded — the abc conjecture — or unbounded for some $\lambda<1$ .

Background

For an integer $n\ge 1$ , its radical $\operatorname{rad}(n)=\prod_{p\mid n}p$ is the product of its distinct prime factors. The abc conjecture (Masser, Oesterlé, 1985) asserts that for every $\varepsilon>0$ there is $K_\varepsilon>0$ such that every triple of coprime positive integers with $a+b=c$ satisfies $\operatorname{rad}(abc)>K_\varepsilon c^{1-\varepsilon}$ ; equivalently, only finitely many coprime solutions violate $\operatorname{rad}(abc)\ge c^{1-\varepsilon}$ . A standard quantitative weakening counts the failures instead of excluding them: for $\lambda\in(0,1)$ , let $N_\lambda(X)$ denote the number of triples $(a,b,c)\in[1,X]^3$ of coprime integers with $a+b=c$ and $\operatorname{rad}(abc)<c^\lambda$ (for fixed $\varepsilon>0$ and $\lambda=1-\varepsilon$ this is the exceptional set $\mathcal{E}(N)$ of the conjecture). The abc conjecture is equivalent to $N_\lambda(X)\ll_\lambda 1$ for every $\lambda<1$ . Since there are $\asymp X^2$ coprime triples in the box, any bound $N_\lambda(X)\ll X^{2-\delta}$ shows the abc conjecture holds for almost all triples. A 1962 estimate of de Bruijn on how many integers $n\le x$ can share a small radical, $|\{n\le x:\operatorname{rad}(n)\le x^\lambda\}|=x^{\lambda+O(1/\sqrt{\log x\log\log x})}$ , yields the folklore bound $N_\lambda(X)\ll_\varepsilon X^{2\lambda/3+\varepsilon}$ , recorded expositively by Lichtman (2025) as $|\mathcal{E}(N)|\ll N^{2/3}$ . For $\lambda>1$ , Mazur (2000) asked whether $N_\lambda(X)$ has exact order $X^{\lambda-1}$ ; Kane (2015) confirmed this for $\lambda\ge 2$ via the determinant method (a technique for bounding integer points on curves and varieties through determinants of interpolation polynomials). For $\lambda<1$ , nothing beyond the folklore bound was known until Browning, Lichtman and Teräväinen (2024) proved $|\mathcal{E}(N)|\ll N^{33/50}$ — the first power saving over $2/3$ since 1962 — by combining geometry of numbers (the study of lattice points in convex bodies), Fourier analysis, Thue equations (classical Diophantine equations $F(x,y)=m$ with irreducible binary forms $F$ of degree $\ge 3$ , which have finitely many solutions by Thue's theorem) and the determinant method to bound the density of integer points on certain high-dimensional varieties. Bernert (June 2025) improved and simplified this argument, and Runbo Li (June 2025), apparently independently, obtained $N_\lambda(X)\ll X^{56/85+\varepsilon}$ ( $56/85\approx 0.6588$ ) by optimizing the original methods; Bernert's improvement was then merged into the four-author paper of Bernert, Browning, Lichtman and Teräväinen (revision of May 2026) proving $N_\lambda(X)\ll_{\varepsilon,\lambda}X^{0.6+\varepsilon}$ for $\lambda\in(0,1)$ , together with the uniform bound $N_\lambda(X)\ll_\varepsilon X^{(23\lambda+3)/40+\varepsilon}$ for $\lambda\in(0,2]$ . The exponents arise from a linear program over parameters that trade off the various density estimates. In a different direction, the best results for individual triples rather than counting functions are due to Stewart and Yu (2001), who showed only finitely many coprime triples satisfy $\operatorname{rad}(abc)<(\log c)^{3-\varepsilon}$ , and Pasten (2024), who reached the subexponential threshold $\exp((\log\log c)^{2-\varepsilon})$ in the range $a<c^{1-\varepsilon}$ . Robert, Stewart and Tenenbaum (2014) proposed a refined quantitative form of the conjecture, predicting that the correct threshold has the shape $\operatorname{rad}(abc)>c^{1-O(1/\sqrt{\log c\log\log c})}$ ; this refined form also remains unproved unconditionally, and the authors of the 2026 merged paper state that they expect a serious numerical improvement of the counting exponent to require new ideas.

Problem Statement

Let $\operatorname{rad}(n)=\prod_{p\mid n}p$ denote the radical of a positive integer. For $\lambda\in(0,1)$ , let

N_\lambda(X)=\#\{(a,b,c)\in[1,X]^3:\ \gcd(a,b)=1,\ a+b=c,\ \operatorname{rad}(abc)<c^\lambda\}

be the counting function of abc triples of exponent $\lambda$ ; for fixed $\varepsilon>0$ and $\lambda=1-\varepsilon$ this is the exceptional set $\mathcal{E}(X)$ of the abc conjecture. The problem is to determine the true order of $N_\lambda(X)$ for $\lambda<1$ , with the following concrete targets: (1) prove a power-saving bound $N_\lambda(X)\ll_{\lambda,\varepsilon}X^{\theta+\varepsilon}$ that holds for every fixed $\lambda\in(0,1)$ with a single absolute constant $\theta<3/5$ (the current best such exponent, due to Bernert–Browning–Lichtman–Teräväinen, being $\theta=3/5$ for all fixed $\lambda\in(0,1)$ , with the uniform-in- $\lambda$ bound $X^{(23\lambda+3)/40+\varepsilon}$ ); (2) at the conjectural endpoint, decide whether $N_\lambda(X)\ll_\lambda 1$ for every $\lambda<1$ — a statement equivalent to the abc conjecture of Masser and Oesterlé; or (3) refute this picture by proving $N_\lambda(X)\to\infty$ for some $\lambda<1$ , i.e. exhibiting infinitely many coprime triples with $a+b=c$ and $\operatorname{rad}(abc)<c^\lambda$ .

Targets (2) and (3) are the two branches of the same decision problem, and target (1) is weaker: it bounds $N_\lambda(X)$ without by itself determining the true order of $N_\lambda(X)$ .

References

Jared Duker Lichtman, 'The $abc$ conjecture is true almost always', arXiv:2505.13991 (2025), DOI 10.48550/arXiv.2505.13991, https://arxiv.org/abs/2505.13991
Christian Bernert, Tim Browning, Jared Duker Lichtman, Joni Teräväinen, 'Bounds on the exceptional set in the $abc$ conjecture', arXiv:2410.12234v2 (2024, revised 9 May 2026), DOI 10.48550/arXiv.2410.12234, https://arxiv.org/abs/2410.12234
Christian Bernert, 'The exceptional set in the $abc$ conjecture', arXiv:2506.13364 (2025), DOI 10.48550/arXiv.2506.13364, https://arxiv.org/abs/2506.13364
Runbo Li, 'On the exceptional set in the $abc$ conjecture', arXiv:2507.02885 (2025), DOI 10.48550/arXiv.2507.02885, https://arxiv.org/abs/2507.02885
N. G. de Bruijn, 'On the number of integers $\leqq x$ whose prime factors divide $n$ ', Illinois Journal of Mathematics 6 (1962), 137–141, DOI 10.1215/ijm/1255631814, https://doi.org/10.1215/ijm/1255631814
Olivier Robert, Cameron L. Stewart, Gérald Tenenbaum, 'A refinement of the $abc$ conjecture', Bulletin of the London Mathematical Society 46 (2014), 1156–1166, DOI 10.1112/blms/bdu069, https://doi.org/10.1112/blms/bdu069
C. L. Stewart, Kunrui Yu, 'On the $abc$ conjecture, II', Duke Mathematical Journal 108 (2001), 169–181, DOI 10.1215/s0012-7094-01-10815-6, https://doi.org/10.1215/s0012-7094-01-10815-6
Héctor Pasten, 'The largest prime factor of $n^{2}+1$ and improvements on subexponential $ABC$ ', Inventiones mathematicae 236 (2024), 373–385, DOI 10.1007/s00222-024-01244-6, https://doi.org/10.1007/s00222-024-01244-6
Daniel Kane, 'On the number of ABC solutions with restricted radical sizes', Journal of Number Theory 154 (2015), 32–43, DOI 10.1016/j.jnt.2014.09.021, https://doi.org/10.1016/j.jnt.2014.09.021