Unconditional quantitative bounds for the four-term $abcd$ equation over $\mathbb{Z}$ without size hypotheses, and the subexponential shape of the $\operatorname{rad}(bc)$ bound towards $abc$

For pairwise coprime non-zero integers $x_1,x_2,x_3,x_4$ with $x_1+x_2+x_3+x_4=0$, no unconditional bound — not even an exponential one — is known that controls $H=\max_j|x_j|$ in terms of the radical $R=\operatorname{rad}(x_1x_2x_3x_4)$ without auxiliary hypotheses on the sizes of subsums; the only unconditional result (Pasten–Sepúlveda-Manzo, 2024) assumes some pairwise subsum is at most $H/\exp((\log H)^\tau\log_2^*H)$. The problem asks for an unconditional explicit bound of $H$ in terms of $R$ (of subexponential shape $\log H\le\exp(o(\log R))$ as $R\to\infty$ ideally, but any explicit bound resolves the main gap), with no such size hypothesis, and more generally for the $n$-term conjectures with $n\ge4$. A related intermediate question raised by the same authors is whether the exponential-shape bound $\log c\le\exp(\tau^{-1}\kappa\cdot(\log_3^*\operatorname{rad}(bc)/\log_2^*\operatorname{rad}(bc))\cdot\log\operatorname{rad}(bc))$, which depends only on $\operatorname{rad}(bc)$, can be improved to something similar to the square-root shape of the $\operatorname{rad}(abc)$-based bound; that intermediate concerns three-term equations and retains a size hypothesis on $a$, so it is an input towards the main gap rather than a substitute for it. No resolution of any part of this is known in the literature to date.

Background

For a non-zero integer $n$, its radical $\operatorname{rad}(n)$ is the product of the distinct prime divisors of $n$ (the largest squarefree divisor). The $abc$ conjecture of Masser and Oesterlé asserts that for each $\varepsilon>0$ there is $K_\varepsilon>0$ such that every triple of coprime positive integers with $a+b=c$ satisfies $c\le K_\varepsilon\operatorname{rad}(abc)^{1+\varepsilon}$; equivalently $\log c\ll_\varepsilon\log\operatorname{rad}(abc)$ up to $\varepsilon$. Unconditionally, all known bounds without size restrictions are of exponential type, $\log c\le K_\varepsilon\operatorname{rad}(abc)^{\alpha+\varepsilon}$, the sharpest being $\alpha=1/3$ due to Stewart and Yu via the theory of linear forms in logarithms (explicit lower bounds, due to Matveev and others, for non-vanishing determinants of logarithms of algebraic numbers). Under the size restriction $a\le c^{1-\eta}$, Pasten later proved a subexponential bound of square-root shape, a substantial improvement of an earlier subexponential bound of his: there is an absolute $\kappa>0$ with

$$\log c\le\eta^{-1}\exp\!\left(\kappa\sqrt{(\log\operatorname{rad}(abc))\log_2^*\operatorname{rad}(abc)}\right),$$

where $\log_k$ denotes the $k$-fold iterated logarithm and $\log_k^*(t)=\log_k(t)$ unless this is less than $1$ or undefined, in which case it is set to $1$. This result combines linear forms in logarithms with a modular approach based on Shimura curves (curves obtained as quotients of the upper half-plane by orders in quaternion algebras, realizable as moduli of abelian surfaces), which relates the $abc$-quality of a triple to heights of points on such curves.

Pasten and Sepúlveda-Manzo (2024; published 2025) revisited these estimates and proved a variation whose key feature is that the radical of $a$ does not appear: if $a+b=c$ with $a\le c/\exp((\log c)^\tau\log_2^*c)$ for some $\tau>0$, then

$$\log c\le\exp\!\left(\tau^{-1}\kappa\cdot\frac{\log_3^*\operatorname{rad}(bc)}{\log_2^*\operatorname{rad}(bc)}\cdot\log\operatorname{rad}(bc)\right).$$

This is subexponential in $\operatorname{rad}(bc)$ (the exponent of $\log c$ is $o(\log\operatorname{rad}(bc))$) but still of exponential shape in $\log\operatorname{rad}(bc)$, in contrast to the square-root shape above. The authors state that improving it to the square-root shape with $\operatorname{rad}(bc)$ in place of $\operatorname{rad}(abc)$ "at present seems difficult", because the proof of the $\operatorname{rad}(abc)$-bound crucially uses estimates from the Shimura-curve approach that depend on the full radical $\operatorname{rad}(abc)$.

The motivation is the multi-term generalization. Browkin and Brzeziński proposed the $n$-conjecture: for fixed $n\ge3$ there is $M_n$ such that every zero-sum $x_1+\cdots+x_n=0$ of integers with $\gcd(x_1,\dots,x_n)=1$ and no vanishing proper subsum satisfies $\max_j|x_j|\le\operatorname{rad}(x_1\cdots x_n)^{M_n}$; a strong variant discussed in Browkin's survey replaces the conditions by pairwise coprimality of the $x_j$. The polynomial analogues (function-field versions, via the Mason–Stothers theorem — the polynomial analogue of the $abc$ inequality, asserting that for coprime polynomials with $f+g=h$ the degree of $h$ is bounded by the number of distinct roots of $fgh$ — and its extensions) are known, but over $\mathbb{Z}$ the authors state they are "not aware of any unconditional result (not even exponential!)" for these conjectures for any $n\ge4$. The case $n=4$ is usually called the $abcd$ conjecture. Applying their $\operatorname{rad}(bc)$-bound, Pasten and Sepúlveda-Manzo proved the first unconditional estimate for $n=4$: if some pairwise subsum satisfies $\min_{i<j}|x_i+x_j|\le H/\exp((\log H)^\tau\log_2^*H)$, where $H=\max_j|x_j|$, then $\log H\le\exp(\tau^{-1}\kappa(\log_3^*R/\log_2^*R)\log R)$ with $R=\operatorname{rad}(x_1x_2x_3x_4)$; equivalently $\log H\ll_\varepsilon R^\varepsilon$ for every $\varepsilon>0$. The auxiliary hypothesis that some subsum of two terms is small is essential to their argument, since it reduces the four-term equation to a three-term one to which the $\operatorname{rad}(bc)$-bound applies.

Problem Statement

The overarching objective is unconditional, quantitative, radical-based control of multi-term additive relations over $\mathbb{Z}$, in the generality of the Browkin–Brzeziński $n$-conjectures, with $n=4$ as the first open case. The target is the following, with an explicit (effective) bound:

(Removal of the size hypothesis.) Prove that there is an explicit function $F$ such that every quadruple of pairwise coprime non-zero integers $x_1,x_2,x_3,x_4$ with $x_1+x_2+x_3+x_4=0$ satisfies $H=\max_j|x_j|\le F(\operatorname{rad}(x_1x_2x_3x_4))$, with no hypothesis on the size of any subsum. The desired shape is subexponential, e.g. $\log H\le\exp(o(\log R))$ as $R\to\infty$ with $R=\operatorname{rad}(x_1x_2x_3x_4)$ (this shape is a desideratum, not a requirement), matching the quality of the known bound of Pasten–Sepúlveda-Manzo that currently requires $\min_{i<j}|x_i+x_j|\le H/\exp((\log H)^\tau\log_2^*H)$; however, since no unconditional bound of any shape is currently known for $n\ge4$ without such a hypothesis, any explicit unconditional $F$ resolves the main gap. The same question, with $n$ terms in place of $4$ for general $n\ge4$ and the corresponding gcd/no-vanishing-subsum hypotheses of the $n$-conjecture, is the natural general form; a bound for $n=4$ alone resolves the source question.

The sought bound is unconditional: it must not rely on the $abc$ or $abcd$ conjectures, on Szpiro's conjecture, or on other unproved conjectural inputs. The strong 4-term inequality $H\le R^{M_4}$ for some constant $M_4$, and the corresponding $n$-term inequality, are stronger statements than the bound sought here.

Source-raised related intermediate. The source authors ask whether their $\operatorname{rad}(bc)$-only bound for $a+b=c$ under $a\le c/\exp((\log c)^\tau\log_2^*c)$, namely the exponential-shape bound $\log c\le\exp(\tau^{-1}\kappa(\log_3^*\operatorname{rad}(bc)/\log_2^*\operatorname{rad}(bc))\log\operatorname{rad}(bc))$, can be improved to "something similar to" the square-root-shaped $\operatorname{rad}(abc)$-bound $\log c\le\eta^{-1}\exp(\kappa\sqrt{(\log\operatorname{rad}(abc))\log_2^*\operatorname{rad}(abc)})$, and state that this "at present seems difficult", because the Shimura-curve estimates feeding the square-root bound depend on the full radical $\operatorname{rad}(abc)$; any such improvement must avoid dependence on $\operatorname{rad}(a)$. The source does not fix the exact placement of the $\tau$-dependence or the constant structure in such an improvement. This intermediate concerns three-term equations and retains a size hypothesis on $a$; it may be posed with that hypothesis kept, weakened, or removed, and it leaves the subsum hypothesis of the four-term bound in place.

References

1. H. Pasten, R. Sepúlveda-Manzo, "On the $abc$ and the $abcd$ conjectures", arXiv:2406.05083 (2024); published as "On the abc and the abcd Conjectures", Bulletin of the Brazilian Mathematical Society, New Series 56 (2025), no. 4, DOI 10.1007/s00574-025-00476-1, https://arxiv.org/abs/2406.05083
2. J. Browkin, J. Brzeziński, "Some remarks on the $abc$-conjecture", Mathematics of Computation 62 (1994), no. 206, 931–939, DOI 10.2307/2153551, https://doi.org/10.2307/2153551
3. J. Browkin, "The $abc$-conjecture", in: Number theory (R. P. Bambah, V. C. Dumir, R. J. Hans-Gill, eds.), Trends in Mathematics, Birkhäuser, Basel, 2000, pp. 75–105, DOI 10.1007/978-3-0348-7023-8_5, https://doi.org/10.1007/978-3-0348-7023-8_5 (zbMATH Zbl 0971.11010)
4. H. Pasten, "The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$", Inventiones mathematicae 236 (2024), no. 1, 373–385, DOI 10.1007/s00222-024-01244-6, https://doi.org/10.1007/s00222-024-01244-6
5. H. Pasten, "Shimura curves and the $abc$ conjecture", Journal of Number Theory 254 (2024), 214–335, DOI 10.1016/j.jnt.2023.07.002, https://doi.org/10.1016/j.jnt.2023.07.002
6. C. L. Stewart, K. Yu, "On the abc conjecture, II", Duke Mathematical Journal 108 (2001), no. 1, 169–181, DOI 10.1215/s0012-7094-01-10815-6, https://doi.org/10.1215/s0012-7094-01-10815-6
7. P. Vojta, "A more general $abc$ conjecture", International Mathematics Research Notices 1998, no. 21, 1103–1116, DOI 10.1155/S1073792898000658, https://doi.org/10.1155/S1073792898000658
8. F. Luca, I. Pink, C. Sándor, "On the largest value of the solutions of Erdős' last equation", International Journal of Number Theory 21 (2025), no. 6, 1281–1295, DOI 10.1142/s1793042125500666, https://doi.org/10.1142/s1793042125500666
9. J. Cuevas Barrientos, H. Pasten, "On the greatest prime factor of polynomial values and subexponential Szpiro in families", arXiv:2504.15971 (2025), https://arxiv.org/abs/2504.15971
10. H. Pasten, "Power-saving bounds for Thue–Mahler and Mordell equations", arXiv:2608.23559 (2026), https://arxiv.org/abs/2608.23559
11. R. Zhang, "The $abcd$ conjecture, uniform boundedness, and dynamical systems", arXiv:2206.09725 (2022, revised 2024), https://arxiv.org/abs/2206.09725
12. J. D. Lichtman, "The $abc$ Conjecture is True Almost Always", The American Mathematical Monthly (2026), DOI 10.1080/00029890.2026.2702867, https://doi.org/10.1080/00029890.2026.2702867