claims depot shelf
The abc Conjecture
Formal claims (Lean)
Stated claims
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The central claim is Theorem 1.2: if Γ and Δ are almost disjoint finitely generated multiplicative groups of algebraic numbers, and x1+y1 and x2+y2 are non-zero and multiplicatively dependent with x1,x2∈Γ, y1,y2∈Δ, then, with finitely many exceptions, x1/x2 = y1/y2 is a root of unity. The proof splits by the exponent of the dependence: if the common power has exponent at least 2, the sum is a perfect power and the finiteness of perfect powers in Γ+Δ bounds heights; if the exponent is 1, the sums are either equal up to a root of unity or their product is a root of unity, each handled by classical unit-equation finiteness. The theorem is non-effective in general, but effective when both groups
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Theorem 1.2 states: For any α∈Q^×\{±1} and m≥1, (i) log_{A^{(m)}}(α)∈A^{(m)}\setminus Q^×; (ii) log_{A^{(m)}}(α)≠0∈A^{(m)} assuming the abc-conjecture. If the proof is correct, the Fermat-quotient logarithm of any rational α≠±1, restricted to primes congruent to 1 mod m, is never represented by a non-zero rational number, and under abc it is never the zero element of the restricted ring. The application Theorem 1.3 adds: under abc, deg_A(log_A(α))>2, so these finite logarithms are not quadratic over Q in Rosen's finite algebraic numbers.
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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
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The authors prove that the Diophantine equation P_s(n) = t^m for m > 2 has only the solutions listed in Theorems 1, 2 and 3 when s belongs to the families s = 2k+4 (k=4,6 or prime 5≤k≤97) and s = k+4 (k=9,15 or prime 3≤k≤97). Although a fully unconditional proof is not obtained for all possible solutions, the authors expect no further solutions on the basis of the generalized Riemann hypothesis and the weak effective abc conjecture.
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By measuring abc-triples with quality metrics based on the doubly geometric mean of their prime factors, families of high-quality triples are identified that satisfy asymptotic bounds analogous to the abc-conjecture. Sharp phase transitions are established for families of such metrics within specified parametrizations for smoothness of primes, using heuristics from the Szpiro ratio for associated Frey curves. Algorithms are implemented to determine triples with high qualities in sub-linear runtime.
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Assuming the number-field abc conjecture over Q(√Δ), only finitely many terms U_n in a nondegenerate Lucas sequence with Q = ±1 and positive discriminant Δ have squarefree part supported on a fixed finite set of rational primes. Consequently the equations A y^k = product of U_{n_i} with pairwise coprime indices admit an abc-conditional finite reduction. The paper also records the corresponding statement for general k and notes a primitive-divisor obstruction.
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This paper presents various transcendence results in the ring A. It strengthens earlier results by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. Several examples of naive transcendental numbers not previously in the literature are presented. Irrationality of numbers such as log_A(2) is proven under the ABC conjecture.
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Starting with Ramanujan's famous taxicab problem, the solvability of the equations p^n + q^n = r^n + s^n and, more generally, p_1^{k_1} + … + p_m^{k_m} = 0 among polynomials can be studied by relating them to the polynomial analog of the abc-conjecture.
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In certain special cases the explicit abc conjecture implies that the equation a1!!⋯at!!=n!! has only finitely many nontrivial solutions.
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The paper's central discovery is that, for many pairs (d,c), the n-th iterate f_{d,c}^n(x)-α has at most d irreducible factors in K[x], and this bound holds uniformly for all n≥1. When α=0, the set of exponents d with this property for every c of positive height has positive asymptotic density. The proof leverages the abc conjecture to rule out excessive factorization in the number-field case; over function fields the same bound holds unconditionally. As applications, the author computes the density of prime divisors in certain forward orbits and proves finiteness of integral points in certain backward orbits.
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The mathematical body proves Theorem 1.1: for a number field $K$ over which the abc-conjecture holds, fixed $\alpha\in K$, and $f(x)=x^d+c$ with $d\ge2$, $c\in K$, if $\varphi(d)$ exceeds a constant fraction of $d$, every prime divisor of $d$ exceeds a constant depending on $\alpha$ and $K$, $\alpha$ is not a fixed point of $f$, and both $h(c)$ and $h(c-\alpha)$ are positive, then $f^n(x)-\alpha$ has at most $\tau(d)$ irreducible factors in $K[x]$ for all $n\ge1$. The author presents this as the first eventual-stability result for a large class of unicritical polynomials of non-prime-powered degree with nonzero basepoint, and draws consequences for the density of prime divisors in forward or
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The central claim is Theorem 4: for distinct positive integers $m < n$ with $\operatorname{rad}(m+i)=\operatorname{rad}(n+i)$ for every $i \le k$, one has $k \ll (\log n)^{3/2}/(\log\log n)^{9/2}$. The proof starts from the observation that $m \equiv n \pmod{\operatorname{rad}(n(n+1)\cdots(n+k))}$ and that this modulus is less than $n$, so the product of the radicals of the $n$-block is at most $e^{k\log k+O(k)}n$. Pigeonholing forces two adjacent entries with small radical product, and applying the effective abc inequality $c < \exp(C\operatorname{rad}(abc)^{1/3}(\log\operatorname{rad}(abc))^3)$ to the triple $(1, n+m, n+m+1)$ yields a contradiction for larger $k$. The paper states explicitly that the argument needs the exponent $1/3$ in the abc bound; with the earlier exponent $2/3$ the method gives no bound at all.
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On its own terms, the paper establishes Theorem 1.3: for any $\varepsilon>0$, there exists a constant $\delta=\delta(\varepsilon)>0$ such that for $0<\lambda<1+\delta$, $N_{\lambda}(X)\ll X^{56/85+\varepsilon}$. The key claim is that the existing combinatorial framework—reducing the counting problem to a Diophantine count $B_d(c,X,Y,Z)$ and bounding it by Fourier, geometry, determinant, and Thue inequalities—can be optimized to the exponent $56/85$ without introducing any new counting input. The paper also shows that this is the limiting exponent of that framework: the case analysis terminates exactly when the parameter $k=49/12-23\theta/4$ enters, and the contradiction at (48) is precisely the defining inequality for $k$ at $\theta=56/85$.
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The central discovery claimed is the symbolic identity 3^p(s+1)=1+$2^{{k-1}}$(2·3^p n+d), which generates abc-triples of the form (a,b,c)=(1,$2^{{k-1}}$(2·3^p n+d),3^p(s+1)). The paper establishes that d must satisfy d≡−($2^{{k-1}}$)^{-1} mod 3^p and be odd, so that c is automatically a multiple of 3^p while b is a multiple of $2^{{k-1}}$. Under the additional heuristic condition that 2·3^p n+d is smooth, log rad(abc) stays small while c grows exponentially with p, producing quality q=log c/log rad(abc) above 1. Computational examples in the range p∈{1,…,6}, k∈{1,…,7}, n=0 include the known triples (1,8,9) and (1,80,81) and the new-looking (1,242,243) and (1,512000,512001) with quality about 1.44. The paper positions this as evidence that the residue-constrained identity is a structured, symbolic method for generating radical-minimising candidates.
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The authors prove that the exceptional set of coprime triples a, b, c with a + b = c for which rad(abc) < c^{1-ε} has size bounded by a power strictly smaller than the total number of such triples up to a given height; the power saving is obtained from upper bounds on the density of integer points on certain high-dimensional varieties that arise in the analysis and are controlled by the geometry of numbers and Fourier analysis.
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The paper establishes that the p-rationality of some complex cubic number fields can be determined in terms of the p-divisibility of certain terms of a third-order recurrence sequence associated with the field. Several examples are constructed, and relations to the generalized abc-conjecture are discussed, leading to explicit fields that satisfy Greenberg's Generalized Conjecture.
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Suppose a²(a² + 1) divides b²(b² + 1) with b > a. The paper proves b ≫ a (log a)^{1/8} / (log log a)^{12} without additional assumptions, using analytic number theory techniques. It further obtains the stronger bound b ≫_ε a^{15/14 - ε} assuming the abc conjecture.