Constantin M. Petridi
We prove that for any positive integer c and any s > 0 there are representations of c as a sum a+b of two coprime positive integers a, b, such that the respective radicals are all greater than K(s)R(c)^(1-s)c^2. For the reprasentations in question, this is a stronger result than the abc-conjecture, which postulates that these representations are greater than k(s)c^1/(1+s).
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We prove that the frequency of abc equations c^n = a+b satisfying the strong abc - conjecture is phi(c^n)/2+o(phi(c^n)/2), for n going to infinity.
The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c) with 16 | abc, then the full ABC conjecture holds. We extend that result to show that, for every integer N, the "congruence ABC conjecture" that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.
Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.
On the $abc$ and the $abcd$ conjectures
本文重新审视 $abc$ 猜想的次指数界，利用对数线性型建立新变体，并在适当假设下证明四项 $abcd$ 猜想的无条件次指数界。
The conjecture of Masser-Oesterlé, popularly known as $abc$-conjecture have many consequences. We use an explicit version due to Baker to solve a number of conjectures.
This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.
We prove that for any positive integer c there are at least N(c), $1\leq N(c) < ϕ(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc, where k an absolute constant.
We prove that the $abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $abc$-Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.
Powerful numbers and the ABC-conjecture
本文探讨了ABC猜想在确定强大数位置中的应用，介绍了该猜想及其在数论中的重要性。
The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $ε> 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c > (rad(a b c))^{1+ε}$. If true, the abc conjecture would imply many other famous theorems and conjectures as corollaries. In this paper, we discuss the abc conjecture and find new applications to powerful numbers, which are integers $n$ for which $p^2 | n$ for every prime $p$ such that $p | n$. We answer several questions from an earlier paper on this topic, assumin…
This note imparts heuristic arguments and theorectical evidences that contradict the abc conjecture over the rational numbers. In addition, the rudimentary datails for transforming this problem into the doimain of equidistribution theory are provided.
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.
This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier conjecture. As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions. The original abc conjecture of Masser and Oesterlé corresponds to the Second Main Theorem with truncated counting functions on P^1 for the…
The exceptional set in the abc conjecture
本文聚焦 abc 猜想的例外集规模问题，在 Browning、Lichtman 与 Teräväinen 的既有成果基础上作出改进与简化。
We show that the $abc$ conjecture of Masser-Oesterlé-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.
An extended generating series of the radical of n, involving two variables, leads to an identity in said variables, which proves Bombieri's abc-Conjecture for certain sets of integers.
Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.
In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.
This note outlines a constructive proof of a proposition in Mochizuki's paper "Arithmetic elliptic curves in general position," making a direct use of computable non-critical Belyi maps to effectively reduce the full $abc$-conjecture to a restricted form. Such a reduction means that an effective $abc$-theorem is implied by Theorem 1.10 of Mochizuki's final IUT paper ("Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations").
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumption of the \textit{abc} conjecture for number fields.
We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.
Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where the dependence on S is explicit. The converse statement is also announced in this note. For both results, the main geometric tool is a theorem of G.V. Belyi.
We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c < R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$ where R(n) is the radical of $n$, is for $c\to\infty$ $$N(c)=(1-\varepsilon)\frac{ϕ(c)}{2}+O\Bigl(\frac{ϕ(c)}{2}\Bigr).$$ An analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without a constant factor) will also be proved.
Assuming the abc conjecture with $ε=1/6$, we use elementary methods to show that only finitely many $s$-Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers $s$, $n$, $b$, and $q$ with $s,b \geq 2$ and $n,q \geq 3$ satisfy \[C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.\]
We show that the abc Conjecture implies the Weak Diversity Conjecture of Bilu and Luca.
We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture is equivalent to the statement that for $a+b+c<1$, the number of solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1 \pm ε}.$ We prove this conjecture as long as $a+b+c \geq 2.$
Are the Collatz and abc conjectures related?
本文建立 Collatz 猜想与 abc 猜想的联系：在 abc 猜想成立下改进一类捷径型 Collatz 序列起始整数的下界，并借助 abc-hits 子集、Collatz 迭代与 Wieferich 素数寻找大型三元组。
We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the $abc$-conjecture. For unicritical polynomials of degree at least five, we require only the standard $abc$-conjecture.
The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than c. Triples of numbers satisfying a+b=c are called abc triples if the product of their distinct prime divisors is strictly less than c. We catalog what is known about abc triples, both numerical examples found through computation and infinite familes of examples established theoreticall…
We show that Vojta's conjecture for some rational surfaces is related to the $abc$ conjecture. More specifically, we prove that Vojta's conjecture on these surfaces implies a special case of the $abc$ conjecture, while the $abc$ conjecture implies Vojta's conjecture on these surfaces. Moreover, for similar but different rational surfaces, we prove Vojta's conjecture unconditionally. To prove these results, we use some (possibly new) properties of Farey sequences.
This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof o…
The $abc$ Conjecture Revisited
该研究提出新的abc型猜想，核心围绕函数\(W(x,y)\)，通过相关应用激发并说明其相关性，主要结果表明对于固定的\(y\in\mathbb{N}\)，\(\limsup_{x \to \infty} \frac{W(x,y)\log\log x}{\log x} = 1\)。
Vojta's abc Conjecture for algebraic tori and applications over function fields
本文证明了函数域上代数环面和环面簇的 Vojta 广义 abc 猜想，并应用其研究了 log 一般型簇的 Lang-Vojta 猜想，同时将该方法推广至复数情形以确定显式例外集。
The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields
本文在abc猜想假设下，将非Wieferich素数增长结果推广至数域，证明Ding的结果在虚二次域整数环中除31个例外外对所有基成立。
Ramanujan, the taxicab problem for polynomials, and the abc-conjecture
本文研究多项式中方程 $p^n+q^n=r^n+s^n$ 的可解性，并探讨更一般的 $p_1^{k_1}+\dots+p_m^{k_m}=0$ 方程，关联到abc猜想。
Nous montrons qu'un raffinement du théorème de Siegel sur les points entiers de courbes algébriques impliquerait la conjecture abc de Masser-Oesterlé. Nous formulons une hypothèse "Siegel uniforme" qui est une majoration de la hauteur des points S-entiers de la courbe, en termes du corps de rationalité et de l'ensemble de places S. La validité de l'hypothèse pour une quelconque courbe algébrique de caractéristique d'Euler-Poincaré strictement négative, impliquerait une version de la conjecture abc. Ceci étend aux points S-entiers des résultats précédents de L. Moret-Bailly, et est en quelque s…
A form of refined Roth's theorem and its application to the $abc$-conjecture
本文提出精细Roth定理的一种新形式，并利用该结果证明了abc猜想的一个特殊情形。
On the exceptional set in the $abc$ conjecture
本文利用 Browning、Lichtman 和 Teräväinen 的优化方法，将 $abc$ 猜想例外集中满足 $c \leqslant X$ 的三元组数量上界从 $O(X^{33/50})$ 改进为 $O(X^{56/85+\varepsilon})$。
Gain Bounds for Diagonal Superelliptic Equations under the Strong ABC Conjecture
本文在强ABC猜想下，通过结构性分析，建立了对角超椭圆方程解的功率增益和近似增益的理论界，验证了数值观察结果。
This note studies various ways of measuring the complexity of primitive solutions to the equation $A \,+\, B\, + \,C = 0$ and conjectures relating them. We define the insulator $\mathcal I(A,B,C)$ of a primitive solution as the smallest positive integer $\mathcal I$ such that the primes dividing the product $ABC \cdot \mathcal I$ are exactly those below a given bound. We show that the strong XYZ Conjecture of Lagarias and Soundararajan implies there are only finitely many primitive solutions $(A,B,C)$ with a given insulator.
Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate which numbers other than one can or cannot be written as a difference of harmonic numbers and we look at their connection to the $abc$-conjecture. We find that there are only eleven numbers less than 100 that cannot be written as a difference of harmonic numbers (we call these $ndh$-numbers). The smallest …
We establish a precise correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as w…
ABC Conjecture: $ABC = 2^m p^n q^r$ with Fermat or Mersenne Primes
研究了ABC猜想中关于费马和梅森素数的三元组解的有限性及具体构造。
The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.
We develop a general framework to study Szpiro's conjecture and the $abc$ conjecture by means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us to obtain several unconditional results for these conjectures. We first prove various general results about modular and Shimura curves, including bounds for the Manin constant in the case of additive reduction, a detailed study of maps from Shimura curves to elliptic curves and comparisons between their degrees, and lower bounds for the Petersson norm of integral modular forms on Shimura curves. Our main appl…
While currently the $abc$ conjecture and work towards it remains open or is disputed, at the same time much work has been done on weaker versions, as well as on its generalisation to number fields. Given integers satisfying $a+b=c$, Stewart and Yu were able to give an exponential bound for $\max(a,\,b,\,c)$ in terms of the radical over the integers, while Györy was able to give an exponential bound in the algebraic number field case for the projective height $H_{K}(a,\,b,\,c)$ in terms of the radical for algebraic numbers. We generalise Stewart and Yu's method to give an improvement on Györy's…
The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$
本文结合超越方法与模方法，利用 Shimura 曲线同 $ABC$ 猜想的联系，改进了 $n^2+1$ 最大素因子的下界，并推进了 $ABC$ 猜想的亚指数界。
Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture
作为算术泰希米勒空间系列研究的第四部分，本文沿用望月新一的研究框架，证明了算术泰希米勒空间理论可导出abc猜想的证明，验证了望月新一的相关断言。
For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their discriminants are squarefree.
We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmi…
On Smirnov's approach to the $abc$ conjecture
本文利用带基点幺半群上的代数几何，为 Smirnov 在 $abc$ 猜想方法中引入的紧化谱及射影直线映射提供了内在解释。
From ABC to Effective Roth and Ridout Constants for Cubic Roots
本文基于ABC猜想和Roth定理，推导出立方根的有效Roth和Ridout常数界，并引入逼近增益概念，探讨了ABC猜想的可能解决策略。
Bounds on the exceptional set in the $abc$ conjecture
本文研究 abc 猜想中例外集的大小，通过数论和调和分析方法，获得幂级数级数的界。
We show a weak form of the function field version of Oesterle's abc conjecture. It asserts that, if $B$ is a complex projective connected curve, the number of intersection points, counted without multiplicities, of a fixed divisor $D$ of degree $d>0$ over $B$ with the graph $H$ of a section $h:B\to B\times \bP^1$ to the first projection is at least $(d-2)n-C(B,D)$, where $n$ is the degree of $H$ over $\bP^1$, and $C(D,B)$ a constant depending only on these two data. We show this number is at least $(d-2[\sqrt {d}]).n-C(D,B)$. The constant is ineffective.
On $abc$ triples of the form $(1,c-1,c)$
本文研究形如 $(1,c-1,c)$ 的 $abc$ 三元组，推导出两个一般性结论，可复原现有文献中首项为1的 $abc$ 三元组序列，为 $abc$ 猜想相关研究提供支撑。
The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture
该研究通过构造特定的弱特殊三维流形，证明了弱特殊猜想与orbifold莫尔德猜想矛盾，进而影响abc猜想，并展示了Enriques和K3表面的非可除退化情况。
We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighti…
A New Lower Bound in the $abc$ Conjecture
本文证明了$abc$猜想中存在无限多互质数满足更极端的下界条件，将常数从6.068提升至6.563，并给出了与满秩幺模格中非零向量$\ell_1$范数相关的常数$δ$的替换表达式。
This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.
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