Pith. sign in

REVIEW 1 cited by

The $abc$ conjecture is true almost always

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The number of exceptional coprime triples (a,b,c) with a+b=c≤N and rad(abc)<c^{1−ε} is at most O(N^{2/3}).

arxiv 2505.13991 v1 pith:GEEXPSHO submitted 2025-05-20 math.NT math.AGmath.CO

Jared Duker Lichtman This is my paper · ORCID
classification math.NTmath.AGmath.CO
keywords varepsilonconjecturetriplesalmostclassicalcoprimeestimateintegers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The abc conjecture says that whenever three whole numbers satisfy a+b=c and have no common prime factor, the product of all distinct primes appearing in abc cannot be much smaller than c. This 'radical' measures how multiplicatively poor the triple is. The conjecture is famous because proving it for every triple would imply many other results in number theory.\n\nThis note studies the easier question: how often can the conjecture fail? For numbers up to N there are about N^2 coprime triples with a+b=c. The author shows that among these, at most about N^{2/3} triples can have rad(abc) < c^{1−ε}. Since N^{2/3} is much smaller than N^2, the exceptional triples have density zero: the abc inequality holds for almost all triples, in a quantitative sense.\n\nThe proof uses two ingredients. First, very few integers can share the same set of distinct prime factors, because the number of such integers is at most N^{o(1)}. Second, for an exceptional triple, the three pairwise radicals rad(ab), rad(ac), and rad(bc) multiply to exactly rad(abc)^2, so at least one of them must be small. Counting possible small radicals then gives the N^{2/3} bound. The note is expository; a stronger bound, O(N^{33/50}), is due to Browning, Lichtman, and Teräväinen.
Extended reading notes

Core claim

Theorem 1.1 states |E(N)| ≤ O(N^{2/3}), where E(N) is the set of coprime triples in {1,...,N}^3 satisfying a+b=c and rad(abc)<c^{1−ε}. If correct, this means the abc inequality holds for a density 1 subset of all coprime abc triples in the cube. The paper derives this from Lemma 2.2, a bound on integers sharing a fixed radical, and from the identity rad(ab)rad(ac)rad(bc)=rad(abc)^2.

Load-bearing premise

The counting argument assumes that every exceptional triple has some pair xy in {ab, ac, bc} with rad(xy) ≤ c^{2/3−ε}. This is asserted in Section 3 on p.4 from rad(ab)rad(ac)rad(bc) = rad(abc)^2 < c^{2−2ε}; however the identity only forces rad(xy) ≤ c^{2/3−2ε/3} for at least one pair. The final theorem survives with the corrected exponent, because N^{2/3−2ε/3} is still O(N^{2/3}), but the written implication is false and must be fixed.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No new entities or fitted constants. The proof uses only standard arithmetic facts; the only nontrivial input is de Bruijn's radical-counting estimate, which is re-proved as Lemma 2.2. The false intermediate exponent is a proof error, not an invented mechanism.

assumptions (2)
  • standard math Unique prime factorization of integers
    Used throughout to factor n and to define rad(n); invoked in the definitions in Section 2.
  • standard math Multiplicativity of rad and the identity rad(ab)rad(ac)rad(bc)=rad(abc)^2
    Used in Section 3; valid because a, b, c are pairwise coprime for triples in E(N).

how reviews work

0 comments
Cite this review

Pith. "Pith review of The $abc$ conjecture is true almost always." pith.science (2026). https://pith.science/paper/GEEXPSHO

@misc{pith2026250513991,
  author       = {Pith},
  title        = {Pith review of: The $abc$ conjecture is true almost always},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GEEXPSHO}},
  note         = {Machine review of arXiv:2505.13991}
}
abstract

Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm rad}(abc) > K_\varepsilon\, c^{1-\varepsilon}$, for some constant $K_\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\in\{1,\ldots,N\}^3$ satisfying $a+b=c$ and ${\rm rad}(abc) < c^{1-\varepsilon}$. The proof is elementary and essentially self-contained. Beyond revisiting a classical argument for its own sake, this exposition is aimed to contextualize a new result of Browning, Lichtman, and Ter\"av\"ainen, who prove a refined estimate $O(N^{33/50})$, giving the first power-savings since 1962.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the exceptional set in the $abc$ conjecture

    math.NT 2025-06 reject novelty 4.0 of 10

    A new, slightly smaller exponent, 56/85, is claimed for the exceptional set of abc triples with c at most X, improving the previous 33/50.

Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    T. F. Bloom, J. D. Lichtman, The Bombieri–Pila determinant method. Preprint, 2023. ( arXiv:2312.12890)

    arXiv 2023

  2. [2]

    Browning, J

    T. Browning, J. D. Lichtman, J. Ter¨ av¨ ainen, The exceptional set in the abc conjecture Preprint, 2024. (arXiv:2410.12234)

    arXiv 2024

  3. [3]

    N. G. de Bruijn, On the number of integers ⩽ x whose prime factors divide n. Illinois J. Math. 6 (1962), 137–141

    work page 1962

  4. [4]

    Granville, T

    A. Granville, T. Tucker, It’s as easy as abc, Notices of the AMS 49, (2002) 1224–1231

    work page 2002

  5. [5]

    Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC

    H. Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC. Invent. Math. 236 (2024), 373–385

    2024

  6. [6]

    C. L. Stewart, K. R. Yu, On the abc conjecture. II. Duke Math. J. 108 (2001), 169–181

    work page 2001

  7. [7]

    Robert, C

    O. Robert, C. L. Stewart, G. Tenenbaum, A refinement of the abc conjecture. Bull. London Math. Soc. 46 (2014), 1156–1166. Department of Mathematics, Stanford University, 450 Jane Stanford W ay, Stanford, CA 94305- 2125, USA Email address : j.d.lichtman@stanford.edu

    work page 2014

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.