Sharp constants and second-order terms in the Robert–Stewart–Tenenbaum refinement of the abc conjecture (Conjectures A, B, C)

Robert, Stewart and Tenenbaum (2014) proposed three refinements of the abc conjecture — Conjectures A, B, C — that predict, for coprime positive integers $a$ and $b$ with $c=a+b$ , the maximal size of $c$ as a function of the squarefree kernel $k(abc)$ with an explicit leading constant ( $4\sqrt{3}$ in the exponent), explicit second-order constants (e.g. $C_1=\beta+\varepsilon$ with $\beta=1+\log 3-(13/6)\log 2$ ), and explicit logarithmic exponents, under a heuristic treating $k(a+b)$ as statistically independent of $(k(a),k(b))$ . A decade of subsequent work — exceptional-set bounds by Lichtman and by Bernert–Browning–Lichtman–Teräväinen, improved lower bounds by Bright, computational abc-triple searches by de Weger, and Letendre's July 2026 reformulation — uniformly treats Conjectures A, B, C as open; no later work proves, refutes, or amends their precise constants. The open problem is to prove or disprove Conjectures A, B, C as stated, determining the sharp asymptotic behavior of $c$ in terms of $k(abc)$ , including whether the predicted limit supremum of 48 for the merit of abc-triples is correct.

Background

For coprime positive integers $a$ and $b$ with $c=a+b$ , the multiplicative structure of the triple is captured by the squarefree kernel (radical) $k(n)=\prod_{p\mid n}p$ , the product of the distinct primes dividing $n$ . The classical abc conjecture of Oesterlé and Masser asserts that for every $\varepsilon>0$ there exists a constant $K_\varepsilon$ such that $c<K_\varepsilon k(abc)^{1+\varepsilon}$ for all coprime triples $a+b=c$ ; it is one of the most consequential open problems in number theory, implying effective finiteness results such as an effective version of Faltings's theorem (the Mordell conjecture) and Szpiro's conjecture relating the conductor and discriminant of elliptic curves. Robert, Stewart and Tenenbaum (2014) observed that the strongest unconditional upper bounds, of Stewart–Yu type $c<\exp(A\,k(abc)^{1/3}(\log k(abc))^3)$ , together with the probabilistic distribution of the values $k(a+b)$ for $a$ and $b$ of given size, motivate a much sharper, fully explicit prediction. Writing $\log_2$ and $\log_3$ for the second and third iterates of the logarithm, their Conjecture A states that there are constants $C_1$ and $C_2$ such that every coprime pair satisfies $c<k\exp(4\sqrt{3\log k/\log_2 k}\,(1+\log_3 k/(2\log_2 k)+C_1/\log_2 k))$ with $k=k(abc)$ , while infinitely many pairs achieve the matching lower bound with $C_2$ ; the analysis suggests one may take $C_1=\beta+\varepsilon$ and $C_2=\beta-\varepsilon$ for any $\varepsilon>0$ , where $\beta=1+\log 3-(13/6)\log 2$ , together with the supplementary constraint $\max(C_1,C_2)<\lambda:=1-(1/2)\log(4/3)$ . Conjecture B replaces the exponential by a sharper function involving $F(t)=(6/\pi^2)\sum_{m>1}\min(1,e^{t}/m)/\varpi(m)$ , where $\varpi(m)=\prod_{p\mid m}(p+1)$ , asserting $c<B_0 k F((2/3)\log k)^{3-B_1/\log_2 k}$ with $B_1=\log 4-\varepsilon$ , and a matching lower bound with $B_2=\log 4+\varepsilon$ . Conjecture C refines this to $c\le B_3 k F((2/3)\log k)^3H(k)(\log k)^{11/2+\varepsilon}$ , with the matching lower bound $c>kF((2/3)\log k)^3H(k)/(\log k)^{3/2+\varepsilon}$ for infinitely many pairs, where $H(k)$ is a coprimality correction factor defined implicitly via a system of transcendental equations; the authors note that they did not try to optimize the logarithmic exponents. All three conjectures follow from a single heuristic assumption (Section 4 of their paper): that for coprime $a$ and $b$ , the kernel $k(a+b)$ behaves as a statistically independent random variable distributed like a typical integer of the same size. A concrete consequence concerns the merit of abc-triples: writing $q=\log c/\log k$ for the quality of a triple and $m=(q-1)^2(\log k)(\log_2 k)$ for its merit, Conjecture A implies that the limit supremum of the merit over all abc-triples equals $48=(4\sqrt{3})^2$ . The merit measures how much $c$ exceeds the radical on the logarithmic scale relevant to the conjecture, and record merit triples (the largest known, with merit about 38.67, was found by Ralf Bonse) provide computational tests of the prediction.

Problem Statement

Prove or disprove the Conjectures A, B, C of Robert–Stewart–Tenenbaum (Bulletin of the London Mathematical Society 46 (2014), 1156–1166) as published. The published paper is definitive: Conjecture A is equations (1·5)–(1·6), Conjecture B is equations (2·5)–(2·6), Conjecture C is equations (2·8)–(2·9), and $H(k)$ is the function defined in Section 4 of the paper; the formulas transcribed here are indicative, and in case of any discrepancy between this transcription and the published text, the published text governs. Throughout, $a$ and $b$ are coprime positive integers, $c=a+b$ , $k=k(abc)$ where $k(n)=\prod_{p\mid n}p$ is the squarefree kernel of $n$ , and $\log_2$ and $\log_3$ denote the second and third iterates of $x\mapsto\max(1,\log x)$ , following the paper's convention. (A) Do there exist constants $C_1$ and $C_2$ such that every coprime pair satisfies $c<k\exp(4\sqrt{3\log k/\log_2 k}(1+\log_3 k/(2\log_2 k)+C_1/\log_2 k))$ , and infinitely many coprime pairs satisfy the matching bound $c>k\exp(4\sqrt{3\log k/\log_2 k}(1+\log_3 k/(2\log_2 k)+C_2/\log_2 k))$ ? In particular, is the heuristic choice $C_1=\beta+\varepsilon$ , $C_2=\beta-\varepsilon$ for every $\varepsilon>0$ , with $\beta=1+\log 3-(13/6)\log 2$ , together with the supplementary constraint $\max(C_1,C_2)<\lambda:=1-(1/2)\log(4/3)$ , correct? (B) With $F(t)=(6/\pi^2)\sum_{m>1}\min(1,e^{t}/m)/\varpi(m)$ and $\varpi(m)=\prod_{p\mid m}(p+1)$ , do there exist positive constants $B_0$ , $B_1$ , $B_2$ such that $c<B_0 k F((2/3)\log k)^{3-B_1/\log_2 k}$ holds for all coprime pairs and $c>k F((2/3)\log k)^{3-B_2/\log_2 k}$ holds for infinitely many? In particular, are the heuristic values $B_1=\log 4-\varepsilon$ and $B_2=\log 4+\varepsilon$ correct? (C) With $H(k)$ the coprimality correction factor of Robert–Stewart–Tenenbaum, defined in their Section 4 by a system of transcendental equations and admitting the asymptotic expansion $\log H(k)=-\sqrt{\log k/\log_2 k}(\sum_{1\le j\le J}R_j(\log_3 k)/(\log_2 k)^j+O((\log_3 k/\log_2 k)^{J+1}))$ for every fixed $J$ , with $R_1=8\log 2/\sqrt{3}$ : does there exist, for each $\varepsilon>0$ , a constant $B_3$ such that $c\le B_3 k F((2/3)\log k)^3H(k)(\log k)^{11/2+\varepsilon}$ holds for all coprime pairs, and $c>k F((2/3)\log k)^3H(k)/(\log k)^{3/2+\varepsilon}$ for infinitely many? Can the non-optimized logarithmic exponents $11/2+\varepsilon$ and $3/2+\varepsilon$ be improved or eliminated? Concretely, these conjectures concern the sharp asymptotic relationship between $c$ and $k(abc)$ ; the 'In particular' subquestions above ask whether the predicted constants ( $4\sqrt{3}$ ; $\beta$ and $\lambda$ ; $\log 4$ ) are correct, and Conjecture A carries the implied corollary that the limit supremum, over all abc-triples (coprime triples $a+b=c$ ), of the merit $m=(q-1)^2(\log k)(\log_2 k)$ , where $q=\log c/\log k$ is the quality of the triple, equals 48. The three conjectures are independent: each is a separate question, and settling any one of them, in either direction, settles only that conjecture.

References

Olivier Robert, Cameron L. Stewart, Gérald Tenenbaum, 'A refinement of the abc conjecture', Bulletin of the London Mathematical Society 46 (2014), no. 6, 1156–1166, DOI 10.1112/blms/bdu069, https://doi.org/10.1112/blms/bdu069
Jared Duker Lichtman, 'The abc conjecture is true almost always', American Mathematical Monthly (2026), DOI 10.1080/00029890.2026.2702867, https://doi.org/10.1080/00029890.2026.2702867; preprint 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:2410.12234 (2024, revised May 2026), https://arxiv.org/abs/2410.12234
Patrick Letendre, 'The abc Conjecture Revisited', arXiv:2607.07641 (2026), https://arxiv.org/abs/2607.07641
Curtis Bright, 'A new lower bound in the abc conjecture', Canadian Mathematical Bulletin 67 (2024), no. 2, 369–378, DOI 10.4153/S0008439523000784, https://doi.org/10.4153/S0008439523000784; preprint arXiv:2301.11056, https://arxiv.org/abs/2301.11056
Benne de Weger, 'A lot of fudge around A + B = C', arXiv:2301.03050 (2023), https://arxiv.org/abs/2301.03050
Benne de Weger, 'On the approximation gain for abc-triples', arXiv:2602.08051 (2026), https://arxiv.org/abs/2602.08051