Note on the Exceptional Set in the ABC Conjecture

Abstract: Fix $\varepsilon&gt;0$, let $x&gt;1$ be a large real number and let $\text{rad}(n)=\prod_{p\mid n}p$ be the radical of an integer $n\geq1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd(a,b,c)=1$, such that $c&gt;(\text{rad}(abc))<sup>{1+\varepsilon}$, is called exceptional triple. Recent works have proved that the cardinality $#\mathscr{E}(x)$ of set $\mathscr{E}$ of exceptional triples satisfies $#\mathscr{E}(x)=O(x<sup>{2/3})$. This note proves that the cardinality of the exceptional set $\mathscr{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally.

• The paper proves that abc-exceptional triples exist for all sufficiently large parameters and that their total number is infinite, using smooth integers near large prime powers.
• The construction combines short-interval smooth-number asymptotics with radical bounds to produce triples whose quality indices approach 5/3, surpassing the best-known explicit example.
• The argument depends on frontier estimates and an assumed local distribution result for distinct prime factors, leaving open the rigor of proposed power-law lower bounds and its relation to the abc conjecture.

Overview

This note by N. A. Carella addresses the exceptional set of the $abc$ conjecture. An exceptional triple $(a,b,c)$ of coprime positive integers with $a+b=c$ satisfies $c > \rad(abc)^{1+\varepsilon}$ for a fixed $\varepsilon>0$, where $\rad(n)$ denotes the product of the distinct prime divisors of $n$. Recent work of Bernert–Browning–Lichtman–Teräväinen (Browning et al., 2024) and Lichtman (Lichtman, 20 May 2025) established the quantitative upper bound $\#\mathscr{E}(x)=O(x^{2/3})$ on the number of exceptional triples with $c>x$. The paper's main contribution is qualitative rather than quantitative: it proves unconditionally that exceptional triples exist for all sufficiently large $x$ and that their cardinality tends to infinity, so the union $(a,b,c)$0 over all $(a,b,c)$1 is infinite. This is a strong claim, since an infinite supply of triples violating $(a,b,c)$2 for every fixed $(a,b,c)$3 would falsify the $(a,b,c)$4 conjecture itself; the argument therefore merits close scrutiny of its inputs.

Method: smooth integers in short intervals

The proof strategy is constructive and rests entirely on the modern theory of $(a,b,c)$5-smooth integers (integers whose prime factors are all at most $(a,b,c)$6). The key input is Younis' uniform asymptotic for smooth numbers in short intervals (Younis, 2024), which states that for $(a,b,c)$7,

$(a,b,c)$8

uniformly for $(a,b,c)$9 and $a+b=c$0 above $a+b=c$1, where $a+b=c$2. Specializing to $a+b=c$3, $a+b=c$4, and the smoothness parameter

$a+b=c$5

the paper obtains a corollary asserting that the interval $a+b=c$6 contains $a+b=c$7 distinct $a+b=c$8-smooth integers, where $a+b=c$9 is the Dickman function evaluated at $c > \rad(abc)^{1+\varepsilon}$0. Although $c > \rad(abc)^{1+\varepsilon}$1, its decay is only of order $c > \rad(abc)^{1+\varepsilon}$2, which is negligible against the polynomial factor $c > \rad(abc)^{1+\varepsilon}$3; hence the expected count diverges. A companion result of Sarvagya (Jain, 14 Feb 2025) guarantees at least one such smooth integer under slightly different hypotheses, providing a consistency check on the interval length required.

Bounding the radical

The second ingredient controls the radical of the smooth integer produced. A lemma shows that if a $c > \rad(abc)^{1+\varepsilon}$4-smooth integer $c > \rad(abc)^{1+\varepsilon}$5 has at most $c > \rad(abc)^{1+\varepsilon}$6 distinct prime factors, then

$c > \rad(abc)^{1+\varepsilon}$7

which is $c > \rad(abc)^{1+\varepsilon}$8, i.e., $c > \rad(abc)^{1+\varepsilon}$9. To ensure most smooth integers in the interval satisfy this sparsity condition on $\varepsilon>0$0, the paper invokes the Erdős–Kac theorem globally [EK1940], its smooth-number variant due to Mehdizadeh (Mehdizadeh, 2017) (valid for $\varepsilon>0$1), and Alladi's complementary distribution result [1982]. A Hardy–Ramanujan tail estimate shows that integers with $\varepsilon>0$2 have counting function $\varepsilon>0$3, so they occupy zero density even within the short interval. It should be noted that one supporting lemma concerning the local distribution of $\varepsilon>0$4 over the short interval is stated under an explicit assumption that the normal law holds locally; the global results cited do not by themselves establish this local version, and this is the weakest link in the distributional part of the argument.

Construction of exceptional triples

The central theorem fixes $\varepsilon>0$5 a large prime power with $\varepsilon>0$6 and $\varepsilon>0$7 chosen so that $\varepsilon>0$8. Taking a $\varepsilon>0$9-smooth integer $\rad(n)$0 with few prime factors, the triple is defined by

$\rad(n)$1

Coprimality follows because every prime divisor of $\rad(n)$2 is at most $\rad(n)$3, so $\rad(n)$4 and no prime can divide both $\rad(n)$5 and $\rad(n)$6 or all three entries simultaneously. Estimating the radical via $\rad(n)$7, $\rad(n)$8, and the smooth-number lemma for $\rad(n)$9, the $n$0 inequality would force

$n$1

Since $n$2 while $n$3 and the exponential factor is $n$4, this fails for all sufficiently large $n$5. Each admissible pair $n$6 thus yields an exceptional triple, and varying $n$7 produces infinitely many. As a quantitative byproduct, the quality index $n$8 achievable by this construction approaches $n$9 as $\#\mathscr{E}(x)=O(x^{2/3})$0, exceeding the index $\#\mathscr{E}(x)=O(x^{2/3})$1 of the best-known explicit $\#\mathscr{E}(x)=O(x^{2/3})$2 triple $\#\mathscr{E}(x)=O(x^{2/3})$3; the paper notes that realizing, say, $\#\mathscr{E}(x)=O(x^{2/3})$4 computationally would require factoring-scale searches over 100-digit integers and is currently impractical.

Relation to known bounds and lower-bound speculation

The existence result sits in apparent tension with nothing in the literature—the $\#\mathscr{E}(x)=O(x^{2/3})$5 upper bound of (Browning et al., 2024, Lichtman, 20 May 2025) concerns the count of exceptions, not their nonexistence—and the two results are compatible: the cube $\#\mathscr{E}(x)=O(x^{2/3})$6 contains about $\#\mathscr{E}(x)=O(x^{2/3})$7 coprime triples, of which at most $\#\mathscr{E}(x)=O(x^{2/3})$8 are exceptional. The paper remarks that its method appears to yield a power lower bound $\#\mathscr{E}(x)=O(x^{2/3})$9 for some $c>x$0, though this is presented as informal observation rather than a proved theorem, and it notes that the ultimate goal in this line of work is to drive the upper bound down to $c>x$1, which would be consistent with the $c>x$2 conjecture up to constants. If the lower-bound speculation were confirmed alongside the existing upper bound, it would pin the true order of the exceptional set between polynomial functions—a scenario incompatible with finiteness of exceptions but not with the conjecture's $c>x$3-formulation, which permits infinitely many violations for each fixed $c>x$4 provided the implied constant absorbs them. Care is needed here: the paper's own framing oscillates between "counterexamples to the $c>x$5 conjecture" and compatibility with it, and the reader should distinguish violations of $c>x$6 (infinitely many allowed) from violations of $c>x$7 with uniform $c>x$8 (the actual conjectural statement).

Limitations and open questions

Several caveats attach to the main theorem. First, the argument depends on short-interval smooth-number asymptotics in ranges ($c>x$9 of size $x$0, intervals of length $x$1) that lie at the current frontier; any revision of these estimates would propagate directly to the conclusion. Second, the zero-density claim for high-$x$2 smooth integers inside the short interval relies partly on a local Erdős–Kac principle that the paper assumes rather than proves, although the global Hardy–Ramanujan tail bound suffices for the specific threshold used. Third, the construction requires $x$3 to be a perfect $x$4-th prime power with $x$5, a sparse set of parameters, and the resulting lower bound on $x$6 is not made effective. Open questions left by the paper include whether the heuristic lower bound $x$7 can be rigorously established, whether the exponent $x$8 imposed by the short-interval technology can be reduced, and how the constructed family interacts quantitatively with the $x$9 upper bound.

Conclusion

The note supplies an unconditional infinitude proof for exceptional triples of the $(a,b,c)$00 conjecture, built from three components: Hildebrand–Tenenbach-type smooth number counts, recent short-interval refinements, and classical distribution results for $(a,b,c)$01. Its most striking feature is the constructive nature of the family, with quality indices approaching $(a,b,c)$02. The result's significance hinges on the correctness of the frontier-level smooth-number inputs and on clarifying the relationship between infinitely many $(a,b,c)$03-violations and genuine counterexamples to the conjecture with uniform constant, questions the paper identifies but does not fully resolve.