Improving the Stewart–Yu bound for all abc triples

Abstract.

We prove that $\log c<AR^{1/3}\exp(-\log R/(3000\log\log R))$ for coprime positive integers $a+b=c$, where $R=\operatorname{rad}(abc)$ and $A$ is absolute and effectively computable. The proof combines estimates at maximal prime powers with a growing number of auxiliary primes in a rational logarithmic form.

1. Introduction

For positive coprime integers $a+b=c$, put $R=\operatorname{rad}(abc)$. The $abc$ conjecture predicts $c<K_{\varepsilon}R^{1+\varepsilon}$ for every $\varepsilon>0$. It remains open. Stewart–Yu [5, p. 226] proved $\log c<R^{2/3+C/\log\log R}$ for $c>2$. Their later bound [6, Theorem 1, p. 170] is $\log c\ll R^{1/3}(\log R)^{3}$, effectively. We prove the following.

Theorem 1.

There is an absolute effectively computable $A>0$ such that

(1.1) $$\log c<AR^{1/3}\exp\!\left(-\frac{\log R}{3000\log\log R}\right)$$

for all positive integers $a+b=c$ with $\gcd(a,b)=1$.

Equivalently, $\log c<AR^{1/3-1/(3000\log\log R)}$, where the exponent tends to $1/3$ from below as $R\to\infty$. Except for $(1,1,2)$, one has $R\geq 6$ and $\log\log R>0$. In particular, $\log c\ll R^{1/3}$ effectively. For $R\geq 6$, Pasten [3, Theorem 1.4] bounds $\log c$ by $P_{\min}\exp(\kappa_{\rm P}\sqrt{\log R\log\log R})$. Here $P_{\min}=\min\{P(a),P(b),P(c)\}$, $P(n)$ is the largest prime factor and $P(1)=1$. For $a\leq c^{1-\vartheta}$ and fixed $\vartheta>0$, he bounds it by $\vartheta^{-1}\exp(\kappa_{\rm P}\sqrt{\log R\log\log R})$, with an absolute $\kappa_{\rm P}>0$. These improve the bound in restricted regions. Neither displayed estimate gives an improvement over the Stewart–Yu bound $\log c\ll R^{1/3}(\log R)^{3}$ for all triples in terms of $R$ alone.

Put

(1.2) $$H=\log c,\qquad W=\log R,\qquad L=1+W,\qquad X=R^{1/3}.$$

Let $M_{i},s_{i},\Delta$ be as in Section 3. Let $C_{T}$ be the effective constant in (3.5) and put $C_{P}=100000C_{V}$, with $C_{V}$ defined in (4.1). On $\mathcal{N}$ in (3.1), we use

(1.3) $$H<C_{T}XL^{2}\Delta^{-2/9},\qquad H<C_{P}XL^{10}\exp\!\left(-\frac{W}{156\log W}+\frac{11}{3}\log\Delta\right).$$

The second holds for $W\geq W_{0}$ when $\log\Delta\leq(3/1820)W/\log W$. At this boundary both rates equal $1/2730>1/3000$. Section 6 gives an effective common threshold. Stewart [4, Section 4, (4.13)–(4.14)] introduced the growing auxiliary-prime construction. Section 4 bounds the cofactors $u_{i}$ used in applying this construction to additive triples. For earlier $p$-adic estimates, see [7]. The use of AI tools is described at the end of the paper.

2. Known estimates

All logarithms are natural. Write $\omega(n)$ for the number of distinct prime divisors of $n$, with $\omega(1)=0$, and $\pi(x)$ for the number of primes at most $x$. For reduced $x/y$, put $h(x/y)=\log\max(|x|,|y|)$. Write $T(z)=\max(1,h(z))$ for rational $z$. The valuation satisfies $v_{p}(p)=1$ and $|x|_{p}=p^{-v_{p}(x)}$.

Fact F1 ([6, Theorem 1, p. 170, (5)]).

There is an absolute effectively computable $S\geq 1$ such that

(2.1) $$H\leq SXW^{3}.$$

Fact F2 ([6, Lemma 2, p. 173, and (10), p. 172]).

For rational $p$-units $\alpha_{1},\ldots,\alpha_{n}$, integer exponents $|b_{i}|\leq B_{0}$, $B_{0}\geq 3$, and $\Theta=\prod_{i}\alpha_{i}^{b_{i}}-1\neq 0$,

(2.2) $$v_{p}(\Theta)<\frac{p}{\log p}(c_{9}n)^{2n}\prod_{i}\frac{\max(h(\alpha_{i}),\log p)}{\log p}\log B_{0},$$

where $c_{9}>0$ is absolute and effectively computable.

In $K=\mathbb{Q}(\alpha_{0})$ of [6, Section 2, (10)], the ramification index at $p$ is $1$, including $p=2$; hence $\operatorname{ord}_{\mathfrak{p}}$ on $\mathbb{Q}$ agrees with $v_{p}$. Independence is not required, and a base $-1$ is allowed.

Fact F3 ([1, Proposition 5, p. 517], corrected in [2]).

Put $K_{0}=300(16e)^{8}$. Let $m\geq 1$ and let $\Gamma$ be generated by distinct primes $q_{1},\ldots,q_{m}$ and $-1$. The primes form a basis for $\Gamma/\Gamma_{\rm tors}$. If $\xi\in\Gamma$ and $\xi\neq 1,-1$, then

(2.3) $$-\log|1-\xi| <K_{0}^{m}\log\max(e,h(\xi))\prod_{j}\log q_{j},$$

(2.4) $$-\log|1-\xi|_{p} <K_{0}^{m}\frac{p}{\log p}\log\max(e,ph(\xi))\prod_{j}\log q_{j}.$$

Lemma 2.1 (Specialisation of the group estimate).

Proposition 5 of [1], with the sign corrected in [2], gives (2.3)–(2.4) under the hypotheses of Fact F3.

Proof.

Apply Proposition 5 of [1], with the sign corrected in [2], with $K=\mathbb{Q}$ and the parameter $\alpha=1$ in [1]. The group has positive rank, and $h(\xi)\geq\log 2>0$. Its constant is $48(16e)^{8}$ for $m=1$. For $m\geq 2$, the specialized coefficient $2(m+1)\log\max(e,m)(16e)^{3m+5}$ is at most $4^{m}(16e)^{8m}\leq\bigl{(}8(16e)^{8}\bigr{)}^{m}$, since $m+1\leq 2^{m}$, $\log\max(e,m)\leq 2^{m-1}$ and $3m+5\leq 8m$. The factor for the real place is at most $F=4/\log 2>1$. For $m\geq 2$, $F\bigl{(}8(16e)^{8}\bigr{)}^{m}\leq\bigl{(}(32/\log 2)(16e)^{8}\bigr{)}^{m}<K_{0}^{m}$. For $m=1$, $48F<300$; both comparisons follow from $\log 2>2/3$. ∎

Fact F4 ([4, Lemma 3.1, pp. 301–302]; [8, Main theorem, (1.18), p. 321]).

Let $p\geq 5$, $n\geq 7$, and $\ell=\log p$. For independent rational $p$-units $\alpha_{i}$ and integers $b_{i}$, not all zero, put $B_{e}=\max(2,|b_{1}|,\ldots,|b_{n}|)$. If $\prod_{i}\alpha_{i}^{b_{i}}\neq 1$, then

(2.5) $$\begin{split}v_{p}\left(\prod_{i}\alpha_{i}^{b_{i}}-1\right)<&\ 376\sqrt{n+1}\,[4+\log(n+1)]\left(7e\frac{p-1}{p-2}\right)^{n}\prod_{i}h(\alpha_{i})\\ &\cdot\max\left(p(n/\ell)^{n},e^{n}\ell\right)\max\bigl{(}\log B_{e},5.4n(n+1)\bigr{)}.\end{split}$$

Lemma 2.2 (Specialisation of the growing-rank estimate).

Under the hypotheses of Fact F4, estimate (2.5) follows from its cited sources.

Proof.

Stewart’s derivation uses the constants $1794$ and $5.25$. Yu’s published rational case has $1790$ and $5.84$; with these constants, the bound $G_{1}\leq 5.4n(n+1)$ used here holds for $n\geq 7$. We derive this range directly from Yu [8]. Use [8, (1.9), (1.11), (1.13), (1.28), (1.30)–(1.31)] with $d=f=e_{\mathfrak{p}}=1$ and Yu’s parameters $q_{\rm Y}=2$, $q_{\rm Y}^{u_{\rm Y}}=2$. Relabel a minimum-valuation exponent last, as (1.17) requires. We retain $5.84$ throughout and use the upper bound $1790\leq 1794$. Put $Q_{n}=4+\log(n+1)$ and $M_{e}=\max(\log B_{e},5.4n(n+1))$. Then

$$\frac{G_{1}}{n+1}<4n+6+\log(4n+6)\leq 5.4n\quad(n\geq 7).$$

The last inequality holds at $7$, and its difference increases. Since $h(\alpha_{i})\geq\log 2$ and the auxiliary quantity in [8, (1.15)] equals $\log 2\log 3/\log 6$, the first term of $h^{(1)}$ is at most

$$\log B_{e}+\log(2\log 3/\log 6)<\log B_{e}+\tfrac{1}{4}<1.001M_{e}.$$

Yu’s additional term $\log B^{\circ}$, where $B^{\circ}=\min\{|b_{i}|:b_{i}\neq 0\}$, satisfies $\log B^{\circ}\leq\log B_{e}\leq M_{e}$. Hence $h^{(1)}\leq\max(1.001M_{e},(n+1)\ell)$. The saturated residue-group index $\delta(\boldsymbol{\alpha})$ is at least $1$. Yu’s main theorem uses the heights of the original independent tuple in both Kummer cases; we only enlarge $p/\delta(\boldsymbol{\alpha})$ to $p$. Yu’s first bound is consequently at most

$$\begin{split}&1794\left(7\frac{p-1}{p-2}\right)^{n}\frac{(n+1)^{n+1}}{2n!\ell^{2}}\prod_{i}h(\alpha_{i})\\ &\quad\cdot\max\left(p(n/\ell)^{n},e^{n}\ell\right)\max(Q_{n},\ell)\max(1.001M_{e},(n+1)\ell).\end{split}$$

Use Stirling’s inequality $(n+1)^{n+1}/n!\leq e^{n+1}\sqrt{n+1}/\sqrt{2\pi}$. Also $\max(Q_{n},\ell)/\ell\leq Q_{n}/\log 5$ and $\max(1.001M_{e},(n+1)\ell)/\ell\leq 1.001M_{e}/\log 5$. Finally,

$$\frac{1.001\cdot 1794e}{2\sqrt{2\pi}(\log 5)^{2}}<376.$$

For this comparison one may use $e<2.718282$, $\log 5>1.609437$, and $\sqrt{2\pi}>2.50662$. This gives (2.5) from the published constants for $n\geq 7$. ∎

Put $C_{H}=6+\log S$. By (2.1), if $h(\xi)\leq H$ and $1\leq y\leq R$, then

(2.6) $$\log\max(e,yh(\xi))\leq C_{H}L,$$

since $\log(RH)\leq\log S+4W/3+3\log W\leq C_{H}L$ and $C_{H}L\geq 6$. For fixed $\gamma>0$, define the effective constants

(2.7) $$\begin{split}U_{\gamma}&=\left\lceil\max(1,4/\gamma^{2},2\log K_{0}/\gamma)\right\rceil,\\ C_{\gamma}&=\prod_{\begin{subarray}{c}q<e^{U_{\gamma}}\\ q\ \mathrm{prime}\end{subarray}}\max\left(1,\frac{K_{0}\max(1,\log q)}{q^{\gamma}}\right).\end{split}$$

For every squarefree $J$,

(2.8) $$G(J):=K_{0}^{\omega(J)}\prod_{q\mid J}\log q\leq C_{\gamma}J^{\gamma},\qquad G(1)=1.$$

For $\zeta=\log q\geq U_{\gamma}$, $\log K_{0}\leq\gamma\zeta/2$ and $\log\zeta\leq\sqrt{\zeta}\leq\gamma\zeta/2$. Multiply over the primes dividing $J$.

3. Maximal prime powers

Arrange $a\leq b$. For each nonunit side $i\in\{a,b,c\}$, put

$$M_{i}=\max_{q\mid i}v_{q}(i),\qquad s_{i}=\min\{q:v_{q}(i)=M_{i}\},\qquad u_{i}=i/s_{i}^{M_{i}},\qquad E_{i}=\operatorname{rad}(u_{i}).$$

By definition, $s_{i}\nmid u_{i}$. On $a^{2}>c$, all three sides exceed $\sqrt{c}$. Put

$$x_{i}=\log s_{i},\quad H_{i}=M_{i}x_{i},\quad\Delta=E_{a}E_{b}E_{c},\quad J_{i}=\Delta/E_{i},\quad\Omega=\log\Delta.$$

Consider the range

(3.1) $$\mathcal{N}=\{a^{2}>c,\ s_{i}^{39}>R^{10}\text{ for every }i\}.$$

On this range,

(3.2) $$R=s_{a}s_{b}s_{c}\Delta,\qquad J_{a}J_{b}J_{c}=\Delta^{2},$$

(3.3) $$\frac{10}{39}W<x_{i}<\frac{19}{39}W,\quad\Omega<\frac{9}{39}W<x_{i},\quad\log\operatorname{rad}(i)<\frac{19}{39}W,\quad H_{i}>\frac{5}{19}H.$$

We have $x_{i}\leq W-x_{j}-x_{k}<19W/39$ and $\log\operatorname{rad}(i)\leq W-x_{j}-x_{k}$; summing the three lower bounds gives $\Omega=W-x_{a}-x_{b}-x_{c}<9W/39$. Maximality gives $H_{i}\geq x_{i}\log i/\log\operatorname{rad}(i)>(10/19)\log i>5H/19$. Also $R\geq 30$, $W>3$ and $W>3L/4$. The actual ratios

(3.4) $$\xi_{a}=b/c,\qquad\xi_{b}=a/c,\qquad\xi_{c}=-a/b$$

have height at most $H$ and satisfy $v_{q}(1-\xi_{i})=v_{q}(i)$ for $q\mid i$.

Set

(3.5) $$C_{U}=2C_{H}C_{1/6},\qquad C_{S}=\frac{2C_{H}K_{0}^{2}C_{1/26}}{(\log 2)^{2}},\qquad C_{T}=\frac{741}{50}C_{H}K_{0}^{2}C_{1/6}.$$

If $a^{2}\leq c$, apply (F3) at infinity to $\xi=b/c$. Here $-\log(1-\xi)\geq H/2$. Equations (2.6)–(2.8) give

(3.6) $$H<C_{U}XLe^{-W/6}.$$

This includes unit sides and $a^{2}=c$. If $a^{2}>c$ and some $s_{i}^{39}\leq R^{10}$, choose $p=s_{i}=\min\{s_{a},s_{b},s_{c}\}$; then $p^{39}\leq R^{10}$. Since $\log i\leq M_{i}\log\operatorname{rad}(i)\leq M_{i}W$ and $\log i>H/2$, maximality gives $H<2WH_{i}/\log p$. Apply (F3) to the full off-side prime basis in (3.4):

$$H_{i}<C_{H}K_{0}^{2}C_{1/26}\frac{p}{\log p}Lx_{j}x_{k}J_{i}^{1/26}\leq C_{H}K_{0}^{2}C_{1/26}\frac{p}{\log p}L^{3}R^{1/26}.$$

Since $p\leq R^{10/39}=XR^{-1/13}$,

(3.7) $$H<C_{S}XL^{4}e^{-W/26}.$$

On all of $\mathcal{N}$, (F3) together with (2.8), with $\gamma=1/6$, gives

$$\begin{gathered}\frac{x_{j}x_{k}}{x_{i}}\leq\frac{39}{10}W,\qquad C_{*}=\frac{39}{10}C_{H}K_{0}^{2}C_{1/6},\\ H_{i}<C_{H}K_{0}^{2}C_{1/6}L\frac{s_{i}}{x_{i}}x_{j}x_{k}J_{i}^{1/6}<C_{*}s_{i}L^{2}J_{i}^{1/6}.\end{gathered}$$

Multiply and use (3.2)–(3.3):

(3.8) $$\prod_{i}H_{i}<C_{*}^{3}RL^{6}\Delta^{-2/3},\qquad H<C_{T}XL^{2}\Delta^{-2/9}.$$

There is no restriction on $\Delta$ in this estimate.

4. Cofactors of the maximal prime powers

Assume the triple lies in $\mathcal{N}$. Put $\kappa=\max(1,c_{9})$ and $F(n)=(\kappa n)^{2n}$. Choose an integer $m_{0}\geq 7$ with $\log(m_{0}+1)\geq 8\log\kappa+10$. Define

(4.1) $$C_{F}=\max\left(1,\max_{0\leq m<m_{0}}\frac{F(m+3)}{((m+1)!)^{3}}\right),\qquad C_{V}=\frac{(C_{H}+2)C_{F}}{4(\log 2)^{3}}.$$

Then $C_{V}\geq 1$, since $C_{F}\geq F(3)\geq 729$. For squarefree $J$ with $m=\omega(J)$, one has $J\geq(m+1)!$. Integration gives $\log((m+1)!)\geq(m+1)\log(m+1)-m$. It follows that

$$\begin{split}\log\frac{F(m+3)}{((m+1)!)^{3}}&\leq(3-m)\log(m+1)+2(m+3)\log\kappa+3m+4+\frac{8}{m+1}\\ &\leq m\left[-\tfrac{1}{2}\log(m+1)+4\log\kappa+5\right]\quad(m\geq 7).\end{split}$$

The last expression is nonpositive for $m\geq m_{0}$. The finite maximum handles $m<m_{0}$, including $0$. Consequently,

(4.2) $$F(3+\omega(J))\leq C_{F}J^{3}.$$

Let $l\mid E_{i}$. Represent $\xi_{i}$ using $s_{j},s_{k}$, every prime dividing $J_{i}$, and the sign. These bases are $l$-units. Their coefficients are at most $B_{0}=\max(3,H/\log 2)$, with $\log B_{0}\leq(C_{H}+2)L$. Since $\log l\leq\Omega<x_{j},x_{k}$, the two factors for $s_{j},s_{k}$ in (2.2) have product at most $L^{2}/[4(\log 2)^{2}]$. Each remaining prime contributes

$$\max(1,\log q/\log l)\leq\max(1,\log q/\log 2)<q.$$

The sign contributes $1$. Apply (F2), (4.2) and $l/\log l\leq E_{i}/\log 2$:

(4.3) $$1\leq v_{l}(i)<C_{V}L^{3}E_{i}J_{i}^{4}\leq C_{V}L^{3}\Delta^{4}.$$

Every residual exponent is included, also at $l=2$. For distinct sides $j,k$, this yields

(4.4) $$T(u_{j}/u_{k})\leq C_{V}L^{4}\Delta^{4}.$$

If $u_{j}>1$, $\log u_{j}\leq\max_{l\mid u_{j}}v_{l}(j)\,\Omega$ and $\Omega<W<L$; if $u_{j}=1$, $\log u_{j}=0$. Likewise for $u_{k}$. Coprimality gives $h(u_{j}/u_{k})=\max(\log u_{j},\log u_{k})$; use $C_{V}\geq 1$ for $T(u_{j}/u_{k})$.

5. Auxiliary primes in the local estimate

Lemma 5.1.

Let $p,q,r$ be distinct primes, $\ell=\log p$, and $\Lambda=\log\ell\geq 10^{6}$. Suppose $\ell\leq\log q,\log r\leq 2\ell$. Let $w_{q},w_{r}$ be coprime positive integers with $\gcd(w_{q}w_{r},pqr)=1$ and $\omega(w_{q}w_{r})\leq\ell$. For $1\leq N_{q},N_{r}\leq p^{2}$, $\sigma\in\{1,-1\}$, and a nonzero difference,

(5.1) $$v_{p}(w_{q}q^{N_{q}}-\sigma w_{r}r^{N_{r}})<25200p\ell^{5}T(w_{q}/w_{r})\exp\!\left(-\frac{\ell}{52\log\ell}\right).$$

Proof.

Write $\beta$ for the valuation and $z=(w_{q}/w_{r})q^{N_{q}}r^{-N_{r}}>0$. If $\beta=0$, the result follows. For $\sigma=1$, take $\nu=1$. For $\sigma=-1$, take $\nu=2$; oddness gives $v_{p}(z^{2}-1)=v_{p}(z+1)=\beta$. Hence $\beta=v_{p}(z^{\nu}-1)$. The denominator is a $p$-unit, and $v_{q}(z)=N_{q}>0$ ensures nonvanishing.

Put $n=\lfloor\ell/(52\Lambda)\rfloor$. Then $n\geq 7$ and $n\geq\ell/(104\Lambda)$. Choose auxiliary primes outside $pqrw_{q}w_{r}$. For $\theta=w_{q}/w_{r}\neq 1$, use $n-3$ of them and set

(5.2) $$\alpha_{1} =q/(\varpi_{4}\cdots\varpi_{n}), b_{1} =\nu N_{q},$$

$$\alpha_{2} =1/r, b_{2} =\nu N_{r},$$

$$\alpha_{3} =\theta, b_{3} =\nu,$$

$$\alpha_{i} =\varpi_{i}\ (i\geq 4), b_{i} =\nu N_{q}.$$

If $w_{q}=w_{r}=1$, omit $\theta$ and use $n-2$ auxiliaries. The product is $z^{\nu}$. Take an integer relation $\prod_{i}\alpha_{i}^{z_{i}}=1$. The valuations at $q$ and $r$ give $z_{1}=z_{2}=0$. For $\theta\neq 1$, a prime in its support then gives $z_{3}=0$, and each $\varpi_{i}$ gives $z_{i}=0$ for $i\geq 4$. The tuple without $\theta$ is treated in the same way. All bases are rational $p$-units.

For $m=\lfloor x/2\rfloor$, Legendre’s formula gives

$$\frac{2^{2m}}{2m+1}\leq\binom{2m}{m}\leq(2m)^{\pi(2m)}.$$

Each prime contributes at most $2m$ to the binomial coefficient. Hence $\pi(x)\log x\geq(x-2)\log 2-\log(x+1)$. For $x\geq 1000$, use $\log 2>2/3$ and $x/15\geq 4/3+\log(x+1)$ to obtain $\pi(x)\geq 3x/(5\log x)$. At $x=6\ell\log(6\ell)$,

(5.3) $$\log x\leq 1.001\Lambda,\qquad\pi(x)\geq\frac{3600}{1001}\ell>3\ell.$$

Here $\log 6+\log(\Lambda+\log 6)\leq 3+\log\Lambda\leq 3+\sqrt{\Lambda}/2\leq 0.001\Lambda$. There are at most $3+\ell$ forbidden primes, and $n+3<2\ell$. Therefore $h(\varpi_{i})\leq 1.001\Lambda$ is possible.

The auxiliary denominator has logarithm at most $1.001n\Lambda<\ell/2$. Since $1.001\Lambda\geq 1$, we have $(1.001\Lambda)^{n-3}\leq(1.001\Lambda)^{n}$; the same comparison holds for $n-2$ auxiliaries. Therefore,

(5.4) $$\prod_{i}h(\alpha_{i})\leq 4\ell^{2}T(w_{q}/w_{r})(1.001\Lambda)^{n},\qquad B_{e}\leq 2p^{2},$$

(5.5) $$\max(\log B_{e},5.4n(n+1))\leq 3\ell^{2}.$$

The cofactor-free tuple satisfies the same bounds. Both maximum terms in (F4) are at most $p(n/\ell)^{n}$, since

$$\log\frac{p(n/\ell)^{n}}{e^{n}\ell}\geq\ell-\frac{\ell}{52\Lambda}[1+\log(104\Lambda)]-\Lambda\geq 0.$$

Here $[1+\log(104\Lambda)]/(52\Lambda)<1/2$ and $\ell/2>\Lambda$. For $p\geq 1000$,

$$\frac{7e^{2}(1.001)(p-1)}{52(p-2)}\leq\frac{7(68/25)^{2}(1001/1000)(999/998)}{52}<1.$$

That upper bound equals $155615229/155937500<0.998$. Consequently,

(5.6) $$\left[7e\frac{p-1}{p-2}\frac{n}{\ell}(1.001\Lambda)\right]^{n}\leq e^{-n}\leq e\exp\!\left(-\frac{\ell}{52\Lambda}\right).$$

Finally, $\sqrt{n+1}[4+\log(n+1)]\leq 2\ell$. Insert (5.4)–(5.6) into (F4), and use $752e\cdot 4\cdot 3<25200$. ∎

6. Proof of Theorem 1

Set

(6.1) $$\tau=\frac{3}{1820},\quad\eta=\frac{1}{3000},\quad\rho=\frac{1}{2730},\quad\mu=\rho-\eta=\frac{3}{91000}>\frac{1}{40000}.$$

Define the effective threshold

(6.2) $$\begin{split}t_{0}=\left\lceil\max\left(1000002,\log(100000C_{V}),\log C_{T},\log C_{U},\log C_{S},\log\frac{2S}{\log 2}\right)\right\rceil,\\ W_{0}=e^{t_{0}},\qquad R_{0}=e^{W_{0}}.\end{split}$$

Assume $W\geq W_{0}$ and put $t=\log W$. Then

(6.3) $$t\geq t_{0},\qquad\log L\leq t+1,\qquad W\geq t^{4},\qquad W/t\geq t^{3}.$$

First suppose the triple lies in $\mathcal{N}$ and $\Omega\leq\tau W/t$. Choose the least $s_{i}=p$, with the other primes denoted by $q,r$. The equation is $c-b=a$, $c-a=b$, or $a+b=c$. The local difference in Lemma 5.1 has valuation $\beta=M_{i}$, with cofactors $u_{j},u_{k}$ and the appropriate sign. Their supports are coprime and avoid $pqr$. By (3.3),

(6.4) $$\frac{10W}{39}<\ell\leq\frac{W-\Omega}{3},\qquad\ell\leq\log q,\log r<\frac{19}{10}\ell<2\ell.$$

In particular, $t-2<\log\ell\leq t$, so $\log\ell\geq 10^{6}$. The upper bound for $\ell$ uses $p^{3}\leq s_{a}s_{b}s_{c}=R/\Delta$, since $p$ is the least of the three primes. Also

$$\omega(u_{j}u_{k})\leq\Omega/\log 2\leq 2\tau W/t<\ell.$$

Every complete exponent is at most $(S/\log 2)e^{W/3}W^{3}$. Its logarithm is at most $W/3+4t<20W/39<2\ell$. The middle comparison follows from $4t<7W/39$ in (6.3). Lemma 5.1 applies with exponents at most $p^{2}$. Using (4.4) gives

(6.5) $$H<\frac{19}{5}\ell\beta<100000C_{V}L^{10}\Delta^{4}\exp\!\left(\ell-\frac{\ell}{52\log\ell}\right).$$

The function $g(x)=x-x/(52\log x)$ increases for $\log x\geq 1$. Put $Y=(W-\Omega)/3$ and $\chi=\Omega/W$. Since $t\chi\leq\tau<\log 3$,

$$\frac{3Yt}{W}-\log Y=\log 3-t\chi-\log(1-\chi)>0,\qquad\frac{Y}{\log Y}\geq\frac{W}{3t}.$$

Apply this to (6.5):

(6.6) $$\begin{split}H&<100000C_{V}XL^{10}\exp\!\left(-\frac{W}{156t}+\frac{11}{3}\Omega\right)\\ &\leq 100000C_{V}XL^{10}e^{-\rho W/t},\end{split}$$

because

(6.7) $$\frac{1}{156}-\frac{11}{3}\tau=\frac{2\tau}{9}=\rho.$$

If $\Omega>\tau W/t$, (3.8) gives $H<C_{T}XL^{2}e^{-\rho W/t}$ on $\mathcal{N}$. The equality $\Omega=\tau W/t$ was included in (6.6).

The polynomial factors satisfy

(6.8) $$\log(100000C_{V})+10\log L \leq 11t+10\leq\mu t^{3}\leq\mu W/t,$$

(6.9) $$\log C_{T}+2\log L \leq 3t+2\leq\mu t^{3}\leq\mu W/t.$$

Here $\mu>1/40000$ and $t^{2}/40000>12$. This covers the cofactor polynomial, $\ell^{5}$, the factor $\ell$, and all constants in (6.5). For $a^{2}\leq c$, or $a^{2}>c$ with some $s_{i}^{39}\leq R^{10}$, (6.3) gives

(6.10) $$W/6-\eta W/t \geq W/12\geq 2t+1\geq\log C_{U}+\log L,$$

(6.11) $$W/26-\eta W/t \geq W/52\geq 5t+4\geq\log C_{S}+4\log L.$$

Here $\eta/t\leq 1/52$ and $\eta/t\leq 1/12$. Equations (3.6), (3.7), and (6.8)–(6.11) prove $H<Xe^{-\eta W/\log W}$ for $W\geq W_{0}$.

It remains to cover the finite range. Apart from $(1,1,2)$, at least two distinct primes divide $abc$. Hence $R\geq 6$. Define

(6.12) $$E_{0}=\max\left(\frac{\log 6}{\log\log 6},\frac{W_{0}}{t_{0}}\right),\qquad A=2\max\left(1,SW_{0}^{3}e^{\eta E_{0}}\right).$$

The function $W/\log W$ has a unique minimum at $W=e$. Its maximum on $[\log 6,W_{0}]$ is therefore $E_{0}$. For $6\leq R<R_{0}$, (F1) yields

$$H\leq SXW^{3}\leq SW_{0}^{3}e^{\eta E_{0}}Xe^{-\eta W/\log W}<AXe^{-\eta W/\log W}.$$

For $(1,1,2)$, $H<1<X$ and $\log\log 2<0$. Hence (1.1) also holds there. ∎

After omitting the cofactor-height and polynomial factors, the formal endpoint $\ell=W/3$ in (5.1) gives

$$X\exp\!\left(-\frac{W}{156\log(W/3)}\right)=X\exp\!\left(-\left(\frac{1}{156}+o(1)\right)\frac{W}{\log W}\right)\qquad(W\to\infty).$$

Its leading coefficient is $1/156$, whereas Theorem 1 uses $1/3000$.

Use of AI tools

The author devised the method of proof. Codex (OpenAI, GPT-6 Astra) and Antigravity (Google, Gemini 4 Carbon) carried out numerical checks and ran blind reviews of the manuscript. Claude Code (Anthropic, Claude Fable 5) checked the proofs and the references. The author checked every step of the proof and every reference, and takes full responsibility for the content of the paper.

Statements and declarations

Competing interests. The author has no competing interests.

Funding. No funding was received for this work.