open problem
The abc Conjecture
A deep statement relating the additive and multiplicative structure of integers, with dozens of famous corollaries.
13 machine-reviewed papers whose extracted
claims name this problem. Matching is literal, over the claim graph, so every listing traces to
a sentence in a review.
This is the review hub: papers that name the problem, with the machine's
verdict on each. The claims shelf lists the
individual claims themselves, formal (Lean) or stated.
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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