Status of Letendre's $H$-function strengthening of the $abc$ conjecture: equivalence with $abc$ and truth

In a July 2026 preprint, Patrick Letendre proposed a strengthening of the $abc$ (Oesterlé–Masser) conjecture in which the radical $\gamma(n)$ is replaced by $H(n)=\gamma(n)/(\log\gamma(n))^{\omega(n)}$, where $\omega(n)$ counts the distinct prime factors of $n$. The conjecture implies $abc$, and the paper poses two precise open problems: does the $abc$ conjecture conversely imply this new Conjecture 1, and can one prove Conjecture 1 false? Both, as well as the truth of Conjecture 1 itself, are open.

Background

For a positive integer $n$, write $\gamma(n)=\prod_{p\mid n}p$ for its radical (the product of its distinct prime factors) and $\omega(n)$ for the number of distinct prime factors. The $abc$ conjecture of Oesterlé and Masser asserts that for every $\epsilon>0$ there is a constant $C(\epsilon)$ such that all triples $(a,b,c)$ of positive integers with $\gcd(a,b,c)=1$ and $a+b=c$ satisfy $c<C(\epsilon)\,\gamma(abc)^{1+\epsilon}$; it is one of the central open problems of Diophantine analysis (see Granville and Tucker, 2002, for a survey). Because $H(n)\le\gamma(n)$ for every $n$ that is not a power of $2$ (the excluded powers of $2$ being the case where $\log\gamma(n)=\log 2<1$), any bound of $abc$ type with $H$ in place of $\gamma$ is a strictly stronger pointwise claim for every admissible triple other than $(1,1,2)$.

Two lines of work calibrate how far such bounds can be sharpened. On the lower-bound side, Stewart and Tijdeman (1986), with subsequent refinements by Stewart and Yu (2001), constructed infinitely many coprime triples $a+b=c$ for which $c$ exceeds $\gamma(abc)$ by an exponential factor of order $\exp(\kappa\sqrt{\log\gamma/\log\log\gamma})$ for some fixed constant $\kappa>0$, showing the $1+\epsilon$ exponent cannot be improved. On the upper-bound side, Robert, Stewart and Tenenbaum (2014) proposed a refinement of the shape $c<\gamma\exp(4\sqrt{3\log\gamma/\log\log\gamma}(1+o(1)))$ and showed it essentially optimal within functions of $\gamma$ alone; Bernert, Browning, Lichtman and Teräväinen (2024) recently bounded the size of the exceptional set of triples violating the $abc$ conjecture.

Letendre (2026) explores a different degree of freedom: functions that use not just $\gamma(abc)$ but the number of prime factors of $abc$. Defining

$H(n):=\frac{\gamma(n)}{(\log\gamma(n))^{\omega(n)}},$

he proposes Conjecture 1: for every $\epsilon>0$ there exists $C(\epsilon)$ such that all coprime $a+b=c$ satisfy $c<C(\epsilon)\,H(abc)^{1+\epsilon}$. Since $H(abc)<\gamma(abc)$ for every admissible triple other than $(1,1,2)$, Conjecture 1 implies the $abc$ conjecture (Proposition 1 of the paper). The function $H$ can be extremely small - its minimum over $n$ with $\omega(n)=k$ is $e^{-k(1+o(1))}$ - yet the author adapts the Stewart–Tijdeman lower-bound construction to $H$ and shows (his Proposition 3) that infinitely many coprime triples satisfy $c>H\exp(2\log H/\log\log H)$, strictly below $H^{1+\epsilon}$; thus the classical smooth-number constructions do not refute the conjecture, and the $1+\epsilon$ exponent remains plausible for the $H$-form. The conjecture is motivated by a rarity heuristic (the count of $n\le X$ with $H(n)\le z$ is $\ll zX^{o(1)}$, his Proposition 4), and it has striking conditional consequences: it implies that for each fixed $y$, the maximal order of $\omega$ over runs $x+1,\dots,x+y$ of $y$ consecutive integers is $(1+o(1))\log x/\log\log x$ (the case $y=2$ goes back to Erdős and Nicolas, 1981), and a lower bound for the least solution of Chinese-remainder systems. All of these results are conditional, since the truth of Conjecture 1 is unknown.

Problem Statement

Let $\gamma(n)=\prod_{p\mid n}p$ denote the radical of $n$, $\omega(n)$ the number of distinct prime factors of $n$, and

$H(n)=\frac{\gamma(n)}{(\log\gamma(n))^{\omega(n)}}.$

Letendre's Conjecture 1 asserts: for every $\epsilon>0$ there exists a constant $C(\epsilon)>0$ such that all triples $(a,b,c)$ of positive integers with $\gcd(a,b,c)=1$ and $a+b=c$ satisfy

$c<C(\epsilon)\,H(abc)^{1+\epsilon}.$

Determine the status of this conjecture relative to the $abc$ conjecture, in the two precise forms posed in the source:

1. (Problem 1) Does the $abc$ conjecture - for every $\epsilon>0$, $c<C(\epsilon)\gamma(abc)^{1+\epsilon}$ - imply Conjecture 1? Since Conjecture 1 implies $abc$, this asks whether the two conjectures are equivalent.

2. (Problem 2) Can one prove that Conjecture 1 is false? A valid refutation must exhibit a fixed $\epsilon>0$ and infinitely many (or unboundedly constant-violating) coprime triples $a+b=c$ with $c\ge C\,H(abc)^{1+\epsilon}$ for every constant $C$; a single finite triple cannot refute the conjecture because $C(\epsilon)$ is existentially quantified.

The truth of Conjecture 1 itself is open, and all conditional consequences in the source paper (the maximal order of $\omega$ over runs of consecutive integers, and the least-solution lower bound for Chinese-remainder systems) depend on it. The source's third question - Problem 3, on the 'ideal function' for an $abc$-type conjecture - is not part of this task.

References

1. Patrick Letendre, The $abc$ Conjecture Revisited, arXiv:2607.07641 (2026), doi:10.48550/arXiv.2607.07641, https://arxiv.org/abs/2607.07641
2. 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
3. C. L. Stewart, R. Tijdeman, On the Oesterlé–Masser conjecture, Monatshefte für Mathematik 102 (1986), 251–257, doi:10.1007/BF01294603, https://doi.org/10.1007/BF01294603
4. C. L. Stewart, Kunrui Yu, On the abc conjecture, II, Duke Mathematical Journal 108 (2001), doi:10.1215/s0012-7094-01-10815-6, https://doi.org/10.1215/s0012-7094-01-10815-6
5. Christian Bernert, Tim Browning, Jared Duker Lichtman, Joni Teräväinen, Bounds on the exceptional set in the abc conjecture, arXiv:2410.12234 (2024), doi:10.48550/arXiv.2410.12234, https://arxiv.org/abs/2410.12234
6. Andrew Granville, Thomas J. Tucker, It's as easy as abc, Notices of the American Mathematical Society 49 (2002), no. 10, 1224–1231, https://www.ams.org/notices/200210/fea-granville.pdf
7. P. Erdős, J.-L. Nicolas, Sur la fonction: nombre de facteurs premiers de N, L'Enseignement Mathématique (2) 27 (1981), 3–27, zbMATH 0466.10037, https://zbmath.org/0466.10037