claims depot shelf
The abc Conjecture
Formal claims (Lean)
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The central claim is Theorems 1 and 2: the inequality family has a sharp transition at $\omega$. If $c>\omega$ and $s \geq (\lfloor 2\theta\rfloor+1)(\lfloor 2\theta\rfloor+2)+1$, then for all sufficiently large $R$ one has $N^\tau_{s,\theta}(R,c\sqrt{X}) \gg_{\theta,s} \tau X^{1-\theta}Y^{s-1}$. If $c<\omega$, a construction with $R=sM^\theta+s\theta M^{\theta-1}$ produces infinitely many $R_n\to
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only wh
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only wh
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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 perf
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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+
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On the paper's own terms, the central discovery is Theorem 1: assuming the abc conjecture, $S(X)=O(X^{13/22})$, and consequently $S(X)/\pi(X)\to 0$ as $X\to\infty$; Ramanujan's tau function misses a density-one subset of the primes. The structural reason is that a prime value $|\tau(n)|=\ell$ forces $n=p^{2k}$, and the layers $k=1,2$ translate into integral points on $y^2=x^{11}\pm\ell$ and $u^2=5
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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; ove
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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
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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 i
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Theorem 1 of the paper states that for every λ∈(0,2), N_λ(X), the number of primitive solutions of a+b=c with c≤X and rad(abc)<c^λ, is O_ε($X^{{(23λ+3)/40+ε}}$). In particular N_1(X)≪$X^{{0.65+ε}}$, which is the sharpest known exponent at λ=1 and improves the earlier 0.66. The theorem also gives N_λ(X)=o($X^{{2λ/3}}$) for every λ∈(9/11,3), so the simple bound with exponent 2λ/3 is not optimal acro
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The report's core claim is that there exists a Teichmüller theory of number fields, and that the failure to construct it explicitly is the sole genuine gap in Mochizuki's Inter-Universal Teichmüller Theory. The report locates the source of the theory in p-adic arithmetic itself: algebraically closed perfectoid fields of characteristic zero with prime residue characteristic, one for each prime p, s
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In the paper's own framing, the central discovery is that the height-restricted sum-of-three-cubes problem is a divisor-counting problem in disguise. Writing $A=x-t$, $B=y-t$, $C=z-t$ converts $x^3+y^3+z^3=n$, $x+y+z=t$ into $ABC=(n-t^3)/3$ with $A+B+C=-2t$, so every representation corresponds to a factorization of $(n-t^3)/3$ twisted by the linear condition; the number of such factorizations is a
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The paper's central claim is that for each admissible torsion type $T = C_2 \times C_{2N}$, a recursive map on $abc$ triples produces an infinite family of good Frey curves. Starting from a good triple $P_0 = (a_0,b_0,c_0)$ with $a_0$ even, $b_0 \equiv 1 \pmod 4$, $b_0/a_0 > \theta_T$, and additionally $a_0 \equiv 0 \pmod 3$ when $T = C_2 \times C_6$, the recursion $P_j = (A_T(a_{j-1},b_{j-1}), B_
Stated claims
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The central discovery is that by combining three gap principles—particle-state, wave-state, and large-scale—with a bootstrapping argument, one can eliminate the large-height condition that previously bounded the admissible range of the parameter \(A\) in the inequality. Theorem 1.2 bounds the number of solutions \(\$\beta$\in K^\times\) to the system (1) by an explicit expression in terms of \(n\), \(\varepsilon\), the heights of the \(\alpha_i\), and the parameter \(A\), with no restriction on \(A\) except \(A\ge 2\). Furthermore, the constructive Proposition 5.4 provides a finite grid on the space of median vectors such that every box contains at most \(N-1\) elements of any \(r\)-separated subsequence, leading to an explicit bound that depends on measure-theoretic covering, equicontinuity, and uniform integrability parameters.
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The load-bearing result is Theorem 1: for the non-isotrivial smooth fibration $\phi: X \to Y$ whose fibers are the genus-6 curves built from a genus-2 curve by the Kodaira construction, every rational point $P \in Y(F)$ satisfies $\omega_P^2 \le (36+\varepsilon)\,\delta(F) + 0(d)$, where $\delta(F) = \log|\Delta_F|$ and $d = [F:\mathbb{Q}]$. Because the Deligne pairing makes $\omega_P^2$ a logarithmic height on $Y$ and the cover $Y \to C$ is finite étale, this transfers to the base curve: $h(P) \le (12+\varepsilon)\,\delta(F)/[F:\mathbb{Q}] + \beta(\varepsilon)$, the effective Mordell bound that the introduction calls Theorem A and Section 7 proves as Theorem 2. Following the template of [16], the same inequality yields a weak Szpiro bound $\log|\Delta(E)| \le (144(1+\log 24)+\varepsilon)\log N(E) + \beta_2(\varepsilon)$ for semistable elliptic curves over $\mathbb{Q}$, a uniform weak $abc$ theorem over number fields, $\log H(a,b,c) \le (24+\varepsilon)\log N(a,b,c) + (30+\varepsilon)\log|\Delta_K|/[K:\mathbb{Q}] + \beta_3(\varepsilon)$, and, via the argument of [8], the absence of Siegel zeros for quadratic characters of negative discriminant.
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The central discovery is that the optimal constant Γ_{1,4} in the positive inhomogeneous problem for indefinite quadratic forms of signature (1,4) equals 8. The paper proves that for every real indefinite quadratic form Q of type (1,4) with nonzero determinant D and every c ∈ ℝ⁵, there exists x ∈ ℤ⁵ such that 0 < Q(x+c) ≤ (8|D|)^{1/5}, and that no smaller constant works. The proof shows that after classical reductions to a canonical form, only nine families of parameters require detailed treatment. These are settled by a combination of analytic arguments (Macbeath's lemma, the uniform residue gap bound, a high-quotient closure lemma), exact finite certificates in the form of dyadic covers and Farkas trees, and the ladder lemma—an explicit infinite family of integer witnesses that resolves three isolated failure points (pinch points) where the reduced auxiliary problem genuinely fails to provide a value in its target interval.
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On the paper's own terms, the central discovery is Theorem 2.2: if the ABC conjecture holds, then any rational point of exact period 3 for $f_c$ yields a parameter $t = x_0 + x_1 = a/b$ in lowest terms with $\max\{|a|,|b|,|a+b|\} \le C_\varepsilon \operatorname{rad}(d)^{1+\varepsilon}$, equivalently $h(t) \le (1+\varepsilon)\sum_{p \in S} \log p + O_\varepsilon(1)$ with $\operatorname{den}(c) = d^2$ and $S$ the support of $d$. The engine is a template valid for every exact period $n \ge 2$: the telescoping identity $\prod_j (x_j + x_{j+1}) = 1$ combines with the theorem that $\operatorname{den}(c)$ is a perfect square, so every consecutive-point sum is an $S$-unit, and then any rational $\tau_n$ with $\tau_n, \tau_n+1 \in \mathbb{Z}[S^{-1}]^\times$ has ABC-bounded height (Theorem 3.4). In period 3 the coordinate change $u = x_0 + x_1$, $v = x_0 + x_2$ reduces the whole condition to $v = -1/(1+u)$, and Proposition 7.2 verifies the template's hypothesis by showing $t, t+1 \in \mathbb{Z}[S^{-1}]^\times$, which is the $S$-unit equation $t/(t+1) + 1/(t+1) = 1$. The paper also characterizes the period-3 locus intrinsically: a rational point $(u,v)$ has exact period 3 for the induced map $G$ if and only if $u \notin \{0,-1\}$ and $v = -1/(1+u)$, and the remaining algebraic factor of $u_0u_1u_2 - 1$ is a ghost curve, a nonsingular cubic over $\mathbb{Q}$ that carries no rational period-3 orbit.
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The core discovery is that the exceptional set of embeddings where a symmetric five-term sum of roots of unity is quasipolynomially small (at scale exp(−C(log N)^13)) is itself polylogarithmically small (at most A(log N)^11). This is unconditional. Additionally, under the uniform number-field abc conjecture, every sum of a fixed number of distinct prime-order roots of unity is uniformly bounded below by exp(−ε p), giving subexponential separation.
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On its own terms, the discovery is a complete description of Pasten's Wronskian lattice and a proof that its norm estimate has the best possible coefficient. Writing $g(n)$ for the greatest common divisor of the numbers $n\,v_p(n)/p$ over the primes $p \mid n$, the Wronskians of the derivations additive on $a+b=c$ are exactly the multiples of $\Delta = \gcd(c g_a g_b, b g_a g_c, a g_b g_c)/\gcd(g_a, g_b, g_c)$, an integer divisible by $abc/\operatorname{rad}(abc)$; and the maximum of $|W_\varphi|/\|\varphi\|$ over non-zero $\varphi \in T(a,b)$ equals $ab\sigma$, where $\sigma$ is the minimum of $S(ab)$, $(c/b)S(ac)$, $(c/a)S(bc)$ and $S(n) = \sum_{p \mid n} v_p(n)/p$. The inequality then reads $c \le R\|\psi\|\sigma/\nu$ with $\nu = \Delta R/(abc)$, giving $X = \log c/\log R \le 1 + \varepsilon_\psi$ and, for every presentation, $G_a \le 1 + \varepsilon_\psi$ and $G_p < 3(1 + \varepsilon_\psi)$. The gap $1 + \varepsilon_\psi - X$ splits as $(\log m_\psi + \log(1/\theta_\psi))/\log R$ into a divisibility factor $m_\psi$ and a weight factor $\theta_\psi$, so the bound is structurally exact; for the triples $1+80=81$ and $1+2400=2401$ it is exactly an equality.
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Let $K = C(B)$ and let $A$ be an abelian variety over $K$. Theorem 1.1 asserts that for any closed subscheme $Y \subset A$ of codimension $> 1$ and any $\varepsilon > 0$, the GCD height satisfies $h_Y(x) \le \varepsilon h_L(x) + c$ for all $x \in A(K)$ outside a proper Zariski closed subset; Theorem 1.3 asserts that for any reduced effective divisor $D$ and finite $S \subset B(\mathbb{C})$, the truncated counting function obeys $N^{(1)}_{D,S}(x) \ge (1-\varepsilon) h_D(x) - c$ outside a proper Zariski closed subset; and Proposition 1.4 upgrades this to $N^{(1)}_{D,S}(x) \ge c_1 h_D(x) - c_2$ with no exceptional set at all. The engine behind all three statements is Proposition 1.11: along any generic sequence of rational points, the discrete proximity, truncated counting, and GCD functions are sandwiched between canonical Xie–Yuan heights $\hat{h}_{D^{\mathrm{sym}},U}$ up to multiplicative factors $(1 \pm \varepsilon)$. The main theorems also show that the exceptional set can be chosen geometrically and is therefore stable under finite field extensions (Theorem 4.13).
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The central discovery is a sharp Roth-type inequality for integral points on the weighted projective line: for any algebraic target α with Φ(α) not on the coordinate axes, the weighted chordal distance δ satisfies -log δ(P,α) ≤ (q0+q1+ε)s(P) + O(1) for all but finitely many integral points P, assuming the abc conjecture. The exponent q0+q1 is optimal when q=(2,3) and Φ(α)=[1:1], where it reduces to Hall's conjecture for $x^{3}$ - $y^{2}$. The proof combines the weighted Veronese map (which expresses the weighted height as 1/L of the ordinary height) with Belyi's theorem and the abc conjecture: integrality forces the numerator and denominator of Φ(P) to be perfect powers, so their radicals are small, and abc controls the size of radical(G(u,v)). The paper also introduces a weighted abc conjecture and weighted Belyi maps to show that the full abc conjecture is equivalent to the weighted conjecture for a single signature.
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On the paper's own terms, the discovery is that two long-standing conjectures and two heuristic gain quantities live in one plane, and the implications between them reduce to the balance ratio. With coordinates (X, Y) and λ = Y/X confined to the limiting range [2, 3] (Lemma 2.2), the abc conjecture becomes the vertical threshold lim sup X ≤ 1 and Szpiro's conjecture for Frey curves the horizontal threshold lim sup Y ≤ 3, with abc implying Szpiro and Szpiro implying lim sup X ≤ 3/2. The gains of Müller–Taktikos and de Weger lie on hyperbolas GaGp = X inside this plane; relative to a presentation P, the identities GaGp = X, fGp = Y and λGa = f hold for the exponent factor f (Proposition 4.2). Because the largest f over presentations equals Y, Szpiro's conjecture is equivalent to the uniform asymptotic bound f ≤ 3 on the exponent factor (Corollary 4.3). The sharpest new claim, Proposition 5.1, is conditional on two unproven bounds (A) lim sup Ga ≤ 3/2 and (B) lim sup Gp ≤ 3 for a fixed presentation rule, and concludes lim sup X ≤ 9/2 and lim sup Y ≤ 27/2. The authors stress that these constants are just products of the separate bounds, well above the thresholds 1 and 3, and that stronger conclusions depend on the presentation or on a joint bound on the gains.
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The central claim is stated as Theorem 1.4: Conjecture 1.1 (Vojta's bounded-degree inequality with discriminant term), Conjecture 1.2 (the admissible-pairs inequality for $(D_0,S)$-integral points), and Conjecture 1.3 (its truncated form) are pairwise logically equivalent. The genuinely new step is the harder implication, Conjecture 1.2 implying Conjecture 1.3, which is established by adapting Vojta's ramified-cover approach to the abc conjecture: one pulls back through a cover where all components of $\pi^*D_1$ have multiplicity at least $e$, uses the discriminant-transformation inequality of Lemma 3.7, and matches the truncated counting function on $X$ against the full counting function on $X'$. The paper further proves (Theorems 1.7 and 5.3) that, conditionally on Conjecture 1.1, certain general-type nonsingular Deligne-Mumford stacks with projective coarse moduli space have a non-Zariski-dense set of $(D_0,S)$-integral points of bounded degree, a form of the Bombieri-Lang conjecture for stacks.
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The central claim is that weak Hall's conjecture is not a separate Diophantine statement but a bound on a single ratio. For a solution $y^2 = x^3 + k$, the paper views $x^3$, $|k|$, $y^2$ as a coprime abc-triple when $\gcd(x,y)=1$, and factors its quality $X = \log M / \log N$, with $M = \max\{x^3, y^2\}$ and $N = \operatorname{rad}(xy|k|)$, through the product $P = xy|k|$ into an approximation gain $G_a = \log M / \log P$ and a power gain $G_p = \log P / \log N \ge 1$, so that $X = G_a G_p$. Theorem 3.1 states that Conjecture 1.1 — finitely many solutions with $|k| < x^{1/2-\delta}$ for every $\delta > 0$ — is equivalent to: only finitely many solutions have $G_a > 1 + \varepsilon$ for every $\varepsilon > 0$. Proposition 3.2 generalizes this: for $6/11 \le \kappa < 6/5$ a gain bound $G_a \le \kappa$ forces $|k| > x^{\gamma(\kappa)-\delta}$ with $\gamma(\kappa) = 3/\kappa - 5/2$, and for $3/4 \le \kappa < 6/5$ the two statements are equivalent. The authors note the lower bound $3/4$ is optimal: the family $(n, 1, 1-n^3)$ has $G_a \to 3/4$, so no two-way dictionary survives below that level.
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The central claim is Theorem 1.8: for every $n\ge 3$, every compact $C^{2n}$ non-degenerate curve $M$ in $\mathbb{R}^n$, and every $\nu>0$, $\Delta>0$, the number $A_M(q,\delta)$ of integral points within distance $\delta$ of the dilated curve $qM$ satisfies $A_M(q,\delta)\ll_{\nu,\Delta}\delta^{n-1}q+q^{n/(2n-1)+\nu}$ for all $q\ge 1$ and $\delta\in[0,\Delta)$, together with the lower bound $A_M(q,\delta)\gg\delta^{n-1}q$ for all sufficiently large $q$ and $\delta\gg q^{-1/(2n-1)+\nu}$. Since $n/(2n-1)=3/5$ at $n=3$, the upper bound coincides with the best previous one there, while for every $n\ge 4$ it is strictly smaller than the exponent $\Theta(n)=(n^2+4)/(n(n+4))$ (even $n$) or $(n^2+3)/(n(n+4)-1)$ (odd $n$) obtained by Hickman and Srivastava, and the $\delta$-threshold for the lower bound is thinner as well. As $\Theta(n)\to 1$ whereas $n/(2n-1)\to 1/2$, the improvement roughly halves the error exponent in high dimensions, leaving a gap of only about $1/(2n)$ to the conjectured optimum $(n-2)/(2n-3)$ proposed by the author. The paper also asserts, in Section 4, that its summation step is optimal within the method, so the remaining gap is attributed to the Fourier-decay and level-set volume estimates rather than to the summing argument.
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The central claim is Theorems 1 and 2: the inequality family has a sharp transition at $\omega$. If $c>\omega$ and $s \geq (\lfloor 2\theta\rfloor+1)(\lfloor 2\theta\rfloor+2)+1$, then for all sufficiently large $R$ one has $N^\tau_{s,\theta}(R,c\sqrt{X}) \gg_{\theta,s} \tau X^{1-\theta}Y^{s-1}$. If $c<\omega$, a construction with $R=sM^\theta+s\theta M^{\theta-1}$ produces infinitely many $R_n\to\infty$ with $N^\tau_{s,\theta}(R_n,c\sqrt{X_n})=0$. The constant $\omega$ emerges from the balance between a linear and a quadratic term in the Taylor expansion of $(X+y)^\theta$ around $X$; this balance is the same mechanism that makes the major-arc integral change sign.
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only when the transport itself carries a positive load. The concrete vehicle is the category of elliptic packages: an elliptic curve presented with a Weierstrass model, local sections, and a label, with a model defect $\mu(f)$ attached to each change of coordinates. The discriminant height obeys $h_\Delta(E')\le h_\Delta(E)+\mu(f)$, making it a transport log-volume, and a dilation tower at the bad prime shows that no isomorphism-invariant scalar can be comparable to it—so keeping the defect visible is not optional. From this the paper derives an intensional Szpiro inequality $h_\Delta(E)\le (6+\varepsilon)n(E)+\delta_{\mathrm{mod}}(E)+C_{\varepsilon,K}$, ties a sharp form to the $abc$ conjecture, and reformulates the Birch and Swinnerton-Dyer conjecture in its pre-modularity form as a statement over finite codes in $\mathsf{ACA}_0$, with the refined constant read as a transport cost separating regulator, period, Tamagawa data, and the order of the Tate–Shafarevich group, whose finiteness is reduced to a single cost hypothesis.
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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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On the paper's own terms, the central discovery is Theorem 1: assuming the abc conjecture, $S(X)=O(X^{13/22})$, and consequently $S(X)/\pi(X)\to 0$ as $X\to\infty$; Ramanujan's tau function misses a density-one subset of the primes. The structural reason is that a prime value $|\tau(n)|=\ell$ forces $n=p^{2k}$, and the layers $k=1,2$ translate into integral points on $y^2=x^{11}\pm\ell$ and $u^2=5x^{22}\pm4\ell$. The paper bounds those point counts under abc and combines them with a known $O(X^{1/2}\log X)$ bound for the $k\ge3$ layer to obtain the stated exponent. A separate heuristic, based on Sato-Tate equidistribution, predicts that the true count is still infinite, of order $X^{1/11}/(\log X)^2$.
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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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Theorem 1 of the paper states that for every λ∈(0,2), N_λ(X), the number of primitive solutions of a+b=c with c≤X and rad(abc)<c^λ, is O_ε($X^{{(23λ+3)/40+ε}}$). In particular N_1(X)≪$X^{{0.65+ε}}$, which is the sharpest known exponent at λ=1 and improves the earlier 0.66. The theorem also gives N_λ(X)=o($X^{{2λ/3}}$) for every λ∈(9/11,3), so the simple bound with exponent 2λ/3 is not optimal across that range. The author presents the result as a demonstration that two of the four methods used in the earlier treatment already suffice to beat the previous record, and sketches a refined version that would lower the λ=1 exponent further.
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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 report's core claim is that there exists a Teichmüller theory of number fields, and that the failure to construct it explicitly is the sole genuine gap in Mochizuki's Inter-Universal Teichmüller Theory. The report locates the source of the theory in p-adic arithmetic itself: algebraically closed perfectoid fields of characteristic zero with prime residue characteristic, one for each prime p, serve as the arbitrary geometric base-points required by Mochizuki's Key Principle of Inter-Universality. These fields give rise to many distinct arithmetic holomorphic structures, so one can compare and average the local and global degree data appearing in central results such as Theorem 3.11 and Corollary 3.12 of Mochizuki's third paper. On this basis the report states, by a point-by-point tabulation, that every mathematical assertion in the Scholze–Stix critique and in Scholze's later review is false, and that the combination of the author's construction with Mochizuki's papers establishes the abc conjecture.
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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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In the paper's own framing, the central discovery is that the height-restricted sum-of-three-cubes problem is a divisor-counting problem in disguise. Writing $A=x-t$, $B=y-t$, $C=z-t$ converts $x^3+y^3+z^3=n$, $x+y+z=t$ into $ABC=(n-t^3)/3$ with $A+B+C=-2t$, so every representation corresponds to a factorization of $(n-t^3)/3$ twisted by the linear condition; the number of such factorizations is a finite sum over divisors, and the paper gives explicit formulas for $R(t,n)$. Using this machinery, the paper proves that no $n\equiv \pm4 \pmod 9$ is representable, and then moves to the height-zero case $t=0$, where representations correspond to positive solutions of $xyz=b_m$ with $x+y=z$. The paper's principal claim is Theorem 9: under a generalized abc conjecture (stated in [1]), the number of positive solutions to $xyz=b_m$ for a suitable sequence $b_m$ tends to infinity, and hence $n=3b_m$ has an unbounded number of representations as a sum of three cubes. If this claim is correct, it would establish that infinitely many integers admit arbitrarily many three-cube representations, a weak form of the long-open conjecture for the full set of admissible $n$.
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Fix $n\ge 2$ and a proper divisor $m$ of $n$, write $t=n/m$, and let $\kappa$ be the squarefree kernel of $m$. The paper proves that, up to an $O(X^{3/4})$ error, the number of $A\le X$ with $A\equiv 0\pmod{\kappa}$ for which $f(x)=x^n+Ax^m+A$ is monogenic with non-squarefree discriminant is $$\frac{X}{\kappa\zeta(2)}\prod_{p\mid\kappa}\left(1-\frac{1}{p+1}\right)\prod_{p\nmid t(t-1)\kappa}\left(1-\frac{1}{$p^{2}$-1}\right).$$ The characterization behind the count is that, under these hypotheses, monogenicity is equivalent to $A$ and $D=t^t+(1-t)^{t-1}A$ both being squarefree. The same squarefree-pair mechanism, with extra coprimality checks limited to the primes $2,3,5,7$, yields the companion count for $A\equiv -1\pmod{\kappa^2}$. A separate construction with $A\ne B$ gives infinitely many $B$ making $x^n+Ax^m+B$ monogenic, unconditional for $2\le t\le 4$ and conditional on the abc conjecture for number fields for $t\ge 5$; when $m\ge2$, infinitely many of those also have non-squarefree discriminant.
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The paper's central claim is that for each admissible torsion type $T = C_2 \times C_{2N}$, a recursive map on $abc$ triples produces an infinite family of good Frey curves. Starting from a good triple $P_0 = (a_0,b_0,c_0)$ with $a_0$ even, $b_0 \equiv 1 \pmod 4$, $b_0/a_0 > \theta_T$, and additionally $a_0 \equiv 0 \pmod 3$ when $T = C_2 \times C_6$, the recursion $P_j = (A_T(a_{j-1},b_{j-1}), B_T(a_{j-1},b_{j-1}), C_T(a_{j-1},b_{j-1}))$ yields, for every $j$, a good $abc$ triple whose induced Frey curve $y^2 = x(x-a_j)(x+b_j)$ is semistable, has torsion subgroup exactly $T$, and satisfies the good-curve inequality $N_E^6 < \max\{|c_4|^3, c_6^2\}$. This establishes Theorem 6.3, the constructive form of the infinitude theorem, uniformly for all four torsion types.
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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.