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CONDITIONAL
Rational points of exact period three for the quadratic family
Assuming ABC, the period-3 parameter t of a rational cycle of x^2 + c satisfies max{|a|,|b|,|a+b|} ≤ Cε rad(d)^{1+ε}, where den(c) = d².
arxiv:2610.39820 · math.NT · 2026-10
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ACCEPT
Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation
For symmetric five-term sums of roots of unity, at most O((log N)^11) of the N^2 possible grid points yield values smaller than exp(-C(log N)^13).
arxiv:2610.27231 · math.NT · 2026-10
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ACCEPT
Roth's theorem on weighted projective lines
The paper proves Roth-type approximation on weighted projective lines, achieving an optimal exponent for integral points under the abc conjecture and unconditionally in certain cases.
arxiv:2610.17589 · math.NT · 2026-10
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CONDITIONAL
The quality plane of abc-triples and the decomposition into gains
A re-coordinatization of abc-triples that makes the known abc-Szpiro implications transparent and yields conditional bounds lim sup X ≤ 9/2 and lim sup Y ≤ 27/2 under two unproven gain hypotheses.
arxiv:2610.03212 · math.NT · 2026-10
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CONDITIONAL
On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
Assuming Vojta's conjecture, three formulations of it, including the admissible-pairs and truncated-counting forms, are proved equivalent, and a Bombieri-Lang-type non-density statement for bounded-degree integral points on certain general type stacks is derived.
arxiv:2609.36300 · math.NT · 2026-09
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REJECT
On the numbers that are sums of three cubes
The paper gives divisor-sum formulas for the number of ordered triples (x,y,z) with x^3+y^3+z^3=n and x+y+z=t, and a conditional, unproven claim that a generalized abc conjecture makes the representation count unbounded for infinitely many n.
arxiv:2009.11972 · math.GM · 2020-09
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CONDITIONAL
Intensional semantics for comparison problems in arithmetic geometry
The paper introduces labeled quantitative (2,1)-categories to make the cost of transporting between isomorphic arithmetic presentations measurable, then reformulates Szpiro, abc, and BSD as transport statements whose arithmetic content remains hypothetical.
arxiv:2608.13609 · math.CT · 2026-08
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REJECT
Final Report on the Mochizuki-Scholze-Stix Controversy
A mathematician claims that Scholze-Stix's objections are entirely false, that Mochizuki's IUTT proof needs his own Arithmetic Teichmuller theory, and that this combination proves the abc conjecture.
arxiv:2505.10568 · math.AG · 2025-04
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ACCEPT
ABC implies that Ramanujan's tau function misses almost all primes
Under the abc conjecture, S(X) = O(X^(13/22)), so the tau function misses almost all primes.
arxiv:2603.29970 · math.NT · 2026-03
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CONDITIONAL
Monogenic trinomials with non-squarefree discriminant
New infinite families and explicit asymptotic densities of monogenic trinomials x^n + A x^m + A with non-squarefree discriminant, plus a conditional construction for A != B.
arxiv:1908.07947 · math.NT · 2019-08
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UNVERDICTED
On the factorization of iterates of x^d+c in large degree
For many degrees d, all iterates of x^d+c factor into at most d irreducible pieces over number fields (assuming abc) and function fields, implying density and finiteness results in arithmetic dynamics.
arxiv:2508.05795 · math.NT · 2025-08
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UNVERDICTED
Guardians and Offenders: A Survey on Harmful Content Generation and Safety Mitigation of LLM
The submission's abstract promises an LLM safety survey, but the provided body is the opening page of an unrelated arithmetic-dynamics paper, so the artifact is internally inconsistent.
arxiv:2508.05775 · cs.CL · 2025-08
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CONDITIONAL
Multiplicative dependence in the sumset of multiplicative groups
Pairs of multiplicatively dependent sums in almost disjoint multiplicative groups are, up to finitely many exceptions, root-of-unity twists; triples are classified assuming the abc conjecture.
arxiv:2607.28857 · math.NT · 2026-07
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CONDITIONAL
The abc Conjecture Revisited
A new abc-type conjecture with H(n)=γ(n)/(logγ(n))^{ω(n)} is proposed; conditional on it, for each fixed y, limsup_{x→∞} W(x,y) loglog x / log x = 1.
arxiv:2607.07641 · math.NT · 2026-07
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ACCEPT
Random Multiplicative Functions and Making Squares from Polynomial Values
Central limit theorems are proved for sums of Rademacher and extended Rademacher random multiplicative functions along polynomial values P(n), by establishing paucity of solutions to P(n₁)P(n₂)P(n₃)P(n₄) = □.
arxiv:2607.06398 · math.NT · 2026-07
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UNVERDICTED
Valuation Separation for Coprime Lucas Products
Establishes valuation separation for products of coprime-index Lucas sequences in Diophantine equations and derives abc-conditional finiteness results for their squarefree parts.
arxiv:2605.24909 · math.NT · 2026-05
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UNVERDICTED
Perfect powers in sequences of polygonal numbers
Extends complete solution lists for perfect-power polygonal numbers to s=2k+4 and s=k+4 (k prime in specified ranges) via modular, hypergeometric, and linear-forms methods plus computations.
arxiv:2606.28227 · math.NT · 2026-06
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UNVERDICTED
Variants on the abc-Conjecture using Alternative Quality Metrics
New quality metrics using doubly geometric means of prime factors in abc-triples produce asymptotic results analogous to the abc-conjecture along with phase transitions and sub-linear algorithms.
arxiv:2606.08416 · math.GM · 2026-06
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UNVERDICTED
Gaps between divisible terms in a² (a² + 1)
Proves b ≫ a (log a)^{1/8} / (log log a)^{12} whenever a²(a²+1) divides b²(b²+1) with b>a, plus a power-saving bound under the abc conjecture.
arxiv:1906.11128 · math.NT · 2019-06
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UNVERDICTED
On p-Rationality of Cubic and Quartic Number Fields
New criterion determines p-rationality of complex cubic fields via p-divisibility in a recurrence sequence, with examples satisfying GGC through generalized abc-conjecture relations.
arxiv:2304.10157 · math.NT · 2023-04
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UNVERDICTED
A comment on the equation n!!=a₁!!cdots a_t!!
The explicit abc conjecture implies only finitely many nontrivial solutions to a1!! ⋯ at!! = n!! in certain special cases.
arxiv:2604.09730 · math.NT · 2026-04
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UNVERDICTED
Ramanujan, the taxicab problem for polynomials, and the abc-conjecture
Solvability of polynomial equations of the form p^n + q^n = r^n + s^n and more general sums of powered polynomials equaling zero is studied, linked to the abc-conjecture.
arxiv:2604.25017 · math.NT · 2026-04
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UNVERDICTED
Some results on naive transcendence in the ring of integers modulo infinitely large primes
Strengthens prior results on naive transcendence in the ring A of integers modulo infinitely large primes and proves irrationality of log_A(2) under the ABC conjecture.
arxiv:2604.25566 · math.NT · 2026-04