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The $abc$ Conjecture Revisited

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proposes a stronger abc-type conjecture based on a log-refined radical, and shows it forces the prime-factor count in short intervals to be asymptotically log x/loglog x.

desk verdict Conjecture 1 is false as stated; the standard abc triple 2 + 3^10*109 = 23^5 gives H(abc) ≈ 1.76 while c ≈ 6.4 million, so no constant C(ε) can work. The conditional theorems are vacuous, though the unconditional sections still have value. read the letter →

arxiv 2607.07641 v2 pith:DQUCGFL5 submitted 2026-07-08 math.NT

Patrick Letendre This is my paper · ORCID
classification math.NT MSC 11N2511N3711N56
keywords abcconjectureradicalindexofcompositionnumberprimefactorsshortintervalsMason–StotherstheoremChineseremainderMersennenumbers
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 paper proposes a strengthened form of the abc conjecture in which the radical is replaced by a smaller quantity, H(abc)=γ(abc)/(log γ(abc))^{ω(abc)}. It argues that H better captures how 'rare' an integer is, and shows that this new conjecture implies the classical abc conjecture. The main result is conditional: assuming the new conjecture, any fixed window of consecutive integers near x contains at most (1+δ) log x / loglog x distinct prime factors in total, for all large x. This makes the previously known lower bound sharp, giving the exact limsup of W(x,y) loglog x / log x = 1. The paper also derives consequences for sums of powers, Mersenne numbers, and systems of congruences.

What carries the argument

The central object is H(n)=γ(n)/(log γ(n))^{ω(n)}: the radical divided by a power of its logarithm, which is much smaller than γ(n) whenever n has many distinct prime factors. The paper motivates Conjecture 1 by a heuristic counting argument (Section 3) showing that triples with unusually small H(abc) are extremely rare. The proofs of Theorems 1 and 2 rely on a family of Mason–Stothers-style polynomial identities P_k(z)=∏_{2|j}(z+j)^{binom{k-1}{j-1}}, Q_k(z) similarly with odd j, and R_k(z)=P_k(z)-Q_k(z). Evaluating these at z=x, their integer values factor in a way controlled by the numbers x+j; applying Conjecture 1 to the reduced identity yields the short-interval bound and the CRT lower

What would settle it

The most direct falsifier would be an explicit infinite family of coprime triples (a,b,c) for which c/H(abc)^{1+ε} → ∞ for some fixed ε>0; the paper's own Mersenne-number example suggests a candidate family. A computation of c/H(abc)^{1.01} on known abc triples with large ratios would immediately show whether violations accumulate, and since Conjecture 1 is stronger than abc, any one such unbounded family would settle the conjecture false regardless of the unspecified constant C(ε).

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Extended reading notes

Core claim

The central claim is Conjecture 1: for every ε>0 there is a constant C(ε) such that every coprime triple a+b=c satisfies c < C(ε) H(abc)^{1+ε}, where H(n)=γ(n)/(log γ(n))^{ω(n)}. Since H(abc) < γ(abc) for every nontrivial triple, this is strictly stronger than the ordinary abc conjecture. The paper's main theorem states that if Conjecture 1 holds, then for every fixed y and δ>0, W(x,y)=Σ_{j≤y}ω(x+j) ≤ (1+δ) log x / loglog x for all x ≥ x_0(δ,y). Consequently, for each fixed y, the limsup of W(x,y) loglog x / log x equals 1, matching a standard lower bound. The proof applies Conjecture 1 to a polynomial identity P_y(x)-Q_y(x)=R_y(x) constructed from binomial coefficients, whose prime factors

Load-bearing premise

Everything rests on the truth of Conjecture 1, an unproven strengthening of abc that the paper supports only with a heuristic and with lower-bound examples, and whose falsity would remove Theorems 1, 2, and Corollary 1.

Editorial extensions

If this is right

  • Conjecture 1 implies the classical abc conjecture, since H(abc) < γ(abc) for all admissible triples except (1,1,2) (Proposition 1).
  • Under Conjecture 1, each fixed window of y consecutive integers has at most (1+δ) log x / loglog x distinct prime factors in total, and this is asymptotically sharp: the limsup equals 1 (Theorem 1 and Corollary 1).
  • For sums of two powers, n=x^a + y^b with gcd(x,y)=1, Conjecture 1 gives ω(n) ≤ (1/a + 1/b + δ) log n / loglog n; similarly ω(n(n+k)) ≤ (1+δ) log n / loglog n for each fixed k.
  • For CRT systems x+a_i ≡ 0 (mod n_i) with pairwise coprime n_i, the least positive solution satisfies x_0 ≥ C(δ,A) (N/H(N))^{1-δ}, where N=n_1⋯n_k (Theorem 2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Conjecture 1 is ever established, it would give a near-tight uniform bound for the total number of prime factors of products of any fixed set of translates, not just single integers—a whole family of 'normal order' statements at once.
  • The Mersenne-number discussion suggests that the true obstruction to a small H(n) is forced arithmetic divisibility (e.g., d|k implies 2^d-1 divides 2^k-1). A weaker but possibly unconditional version of Theorem 1's conclusion might be derived from existing anatomy-of-integers results, since the polynomial construction only needs a bound of the same shape.
  • The paper leaves open whether abc itself implies Conjecture 1 (Problem 1). A natural test is whether the best available refinements of abc can be reshaped into the H-based form; the answer would clarify whether Conjecture 1 is a genuine strengthening or merely a repackaging of known heuristics.
  • Conjecture 1, if true, would also settle sharp lower bounds for the least solution of CRT systems with prescribed moduli, connecting the 'atypicality' of integers to a single unified measure H.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a new abc-type conjecture, Conjecture 1, which replaces the radical γ(n) in the abc inequality by H(n)=γ(n)/(log γ(n))^{ω(n)}. It proves that Conjecture 1 implies the classical abc conjecture, gives a heuristic motivation for Conjecture 1, and derives several conditional consequences. The main result (Theorem 1) states that, assuming Conjecture 1, for each fixed y the sum W(x,y)=Σ_{j=1}^y ω(x+j) satisfies W(x,y)≤(1+δ) log x / log log x for all sufficiently large x. Theorem 2 gives an analogous lower bound for the least solution of a system of congruences, and Proposition 3 constructs infinitely many abc triples with c>H exp(2 log H / log log H).

Significance. The conjectured strengthening is natural and, if true, would give a sharp upper bound for the number of prime factors in short intervals — a result not known to follow from the classical abc conjecture. The polynomial constructions (Lemmas 5 and 6) and the conditional proofs are interesting and mostly transparent. The paper is honest about the conjectural nature: all main theorems are conditional on Conjecture 1, which is strictly stronger than abc and for which no numerical evidence is supplied. The central value is therefore as a conditional contribution, contingent on a new and as yet unsupported conjecture.

major comments (3)
  1. [Remark 1, §7.2 and §7.3] The uniform bound on G(x)=gcd(P(x),Q(x)) is load-bearing for both Theorem 1 and Theorem 2, but the proof is only a sketch. The argument 'bound the largest power of p that can divide at least two of the integers x+a_j' does not by itself control v_p(x+a_j), which is unbounded as x varies. A correct proof should invoke the resultant Res(P,Q) to show G(x) is bounded by a constant depending only on the a_j. Please supply a rigorous proof of (7.4) or replace it with a standard resultant argument. Without this, the proofs of Theorems 1 and 2 contain a gap.
  2. [§3, heuristic for Conjecture 1] The dyadic summation appears to misstate the count. For fixed A with a∈[A,2A), b∈[B,2B), the expected number of triples with H(ab(a+b))<B^{1-ε} is of order B^{-ε}(AB^2)^{o(1)}, and summing over A≤B gives B^{-ε+o(1)} rather than the claimed B^{-ε/2}. The conclusion that the expected number tends to 0 is unchanged, but the displayed estimate should be corrected. This does not invalidate the heuristic, but it should be made accurate.
  3. [§8, Problem 2] The paper explicitly leaves open whether Conjecture 1 is false, and all main theorems collapse if it fails. For a conjecture that is strictly stronger than abc, some numerical verification would substantially strengthen the case for it. I recommend checking Conjecture 1 against known abc triples (e.g., the ABC@Home database) or at least providing a quantitative discussion of the heuristic's reliability for large c. As written, the plausibility of the central conjecture rests entirely on the informal §3 heuristic.
minor comments (6)
  1. [Lemma 6 and throughout] The degree in Lemma 6 should read 2^{k-2}, not '2k−2'; the same superscript appears to be lost in the proof of Theorem 1. Please fix all such exponents.
  2. [Proposition 3] The factorization 'N=ab into two integers of size N^{1/2}(log A)^{O(1)}' is asserted without proof. This is standard but should be justified or given a reference.
  3. [Lemma 5] The notation m_j=... and n_j=... for even/odd j is hard to parse. Please state explicitly that m_j=D_j/e for even j and n_j=0, and vice versa for odd j.
  4. [§4.2] The bound xy≤n^{1/a+1/b} follows from x^a≤n and y^b≤n, but this is not stated. A one-line justification would help.
  5. [§3] The derivation of the expression for n^r and its passage to H(n) is very compressed. Expanding this would improve readability and make the heuristic more convincing.
  6. [Proposition 4] The sentence 'the contribution of the smaller values already satisfies the desired inequality' should be made explicit; the reader is left to reproduce the (straightforward) estimate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are explicitly conditional on the new Conjecture 1, which is not derived from the theorems themselves.

full rationale

The paper's central claim is Conjecture 1, a new strong abc-type statement. All main results (Theorems 1 and 2, Corollary 1) are proved conditionally on Conjecture 1, and the proofs apply Conjecture 1 to auxiliary polynomial identities (Lemma 5/6) rather than assuming the conclusion. Proposition 1 shows Conjecture 1 implies abc, which is not circular since H(n)<γ(n) for nontrivial triples is an independent elementary fact. Section 3 offers only a heuristic for Conjecture 1 and explicitly leaves open whether Conjecture 1 is false (Section 8, Problem 2); this is an honest limitation, not a circular derivation. The lower bound in Corollary 1 uses CRT and PNT. Self-citations ([7], [10]) only introduce auxiliary functions ϑ(n) and related estimates, and the proofs of Lemmas 3 and 4 are given in the paper; they are not load-bearing in the sense of assuming the target result. No fitted parameters are relabeled as predictions, and no self-citation chain is used to force the conclusions. Therefore the paper shows no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The load-bearing assumption is Conjecture 1, an ad-hoc strengthening of abc. Everything else is standard analytic number theory. The heuristic independence in §3 is a background assumption for the conjecture's plausibility, not for the conditional theorems.

assumptions (7)
  • ad hoc to paper Conjecture 1: c < C(ε) H(abc)^{1+ε} for all abc triples.
    Postulated in Section 1; Theorems 1-2 and Corollary 1 are derived from it. Unproven; the paper asks in Section 8 whether it is false.
  • standard math Prime number theorem with explicit error term (π(x)=x/log x + x/log²x + O(x/log³x)).
    Used in Lemma 2, Proposition 3, Corollary 1 lower bound.
  • standard math Hardy-Ramanujan/Lemma B estimate for integers with exactly k prime factors.
    Lemma 1, used in Proposition 4.
  • standard math Mason-Stothers theorem (degree bound for polynomial abc).
    Basis for the polynomial identities P-Q=R in Lemmas 5-6.
  • standard math Chinese remainder theorem existence/size bound for x0<N.
    Used for the lower bound in Corollary 1 and in Theorem 2 setup.
  • standard math Primitive divisor theorem for Mersenne numbers.
    Used in Section 4.2 to lower-bound ω(2^k-1); not load-bearing for main theorems.
  • domain assumption Independence heuristic for counts of integers n=ab(a+b) with small H(n).
    Section 3 motivation for Conjecture 1; assumes the set of rare n and the set of values ab(a+b) have expected intersection size ~ product measure.

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Cite this review

Pith. "Pith review of The $abc$ Conjecture Revisited." pith.science (2026). https://pith.science/paper/DQUCGFL5

@misc{pith2026260707641,
  author       = {Pith},
  title        = {Pith review of: The $abc$ Conjecture Revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQUCGFL5}},
  note         = {Machine review of arXiv:2607.07641}
}
abstract

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function $$ W(x,y) := \sum_{j = 1}^{y}\omega(x+j) \quad (y \in \mathbb{N},\ x \in \mathbb{Z}_{\ge 0}) $$ where $\omega(n)$ denotes the number of distinct prime divisors of $n$. The new conjecture implies that, for each fixed $y \in \mathbb{N}$, $$ \limsup_{x \to \infty} \frac{W(x,y)\log\log x}{\log x} = 1. $$

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Works this paper leans on

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