Abc Conjecture Research Articles

A complex case of Vojta’s general abc conjecture and cases of Campana’s orbifold conjecture

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into P 2 \mathbb P^2 intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map f : C ↦ P n f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n + 1 n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f f . As applications, we study some cases of Campana’s orbifold conjecture for P 2 \mathbb P^2 and finite ramified covers of P 2 \mathbb P^2 with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of G m 2 \mathbb G_m^2 .

On -unramified extensions over imaginary quadratic fields

AbstractLet $n$ be an integer congruent to $0$ or $3$ modulo $4$ . Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$ . The same result is obtained unconditionally in special cases.

A problem of Erdős–Graham–Granville–Selfridge on integral points on hyperelliptic curves

AbstractErdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$ , under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$ , and in the process solve Granville’s problem unconditionally.

Shimura curves and the abc conjecture

In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields.Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L2-norm of integral quaternionic modular forms.The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations.

Deep Disagreement in Mathematics

Disagreements that resist rational resolution, often termed ``deep disagreements'', have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki's work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice.

Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations

The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory” . In the first three papers of the series, we introduced and studied the theory surrounding the log-theta-lattice , a highly noncommutative two-dimensional diagram of “miniature models of conventional scheme theory” , called \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters , that were associated, in the first paper of the series, to certain data, called initial \Theta -data . This data includes an elliptic curve E_F over a number field F , together with a prime number l\ge 5 . Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids” . Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure” , i.e., the ring/scheme structure of a \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater related to a given \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture , and the Szpiro Conjecture for elliptic curves. Finally, we examine – albeit from an extremely naive/non-expert point of view! – the foundational/set-theoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species” , which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object” . These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry , as well as to the idea of gluing together distinct models of conventional scheme theory , i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal” .

Two restricted ABC conjectures

Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums a+b=c such that D|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.

Conjecturing with some Conjectures

This world has been seeing so many conjectures from the formation of this world. Particularly, mathematics world has been seeing so many conjectures from the civilization of this world, also venturing through on it. So many Indians, Chinese and Arabian scholars provided their participation in mathematics world. Since before Euclid so many mathematicians ventured on numbers, geometry, astronomy, etc... in number theory, there are so many conjectures like applications of GCD, Fermat Last theorem, Euler’s totient function, Gold Bach conjecture, ABC conjecture, etc... some of this has been proved but so many of that still has not been proved. In this paper, I try to prove Gold Bach conjecture, stating abc conjecture for composite numbers, and try to deliver some conjectures.

On a generalization of a conjecture of Grosswald

We generalize a conjecture of Grosswald, now a theorem due to Filaseta and Trifonov, stating that the Bessel polynomials, denoted by yn(x), have the associated Galois group Sn over the rationals for each n. We consider generalized Bessel polynomials yn,β(x) which contain interesting families of polynomials whose discriminants are nonzero rational squares. We show that the Galois group associated with yn,β(x) always contains An if β≥0 and n sufficiently large. For β<0 the Galois group almost always contains An. It is further shown that for β<−2, under the hypothesis of the abc conjecture, the Galois group of yn,β(x) contains An for all sufficiently large n. Using these results, an earlier work of Filaseta, Finch and Leidy and the first author concerning the discriminants of yn,β(x), we are able to explicitly describe the instances where the Galois group associated with yn,β(x) is An for all sufficiently large n depending on β.

A Proof Of The ABC Conjecture.

In this article, its shown that the ABC Conjecture is correct for integers a+b=c, and any real number r>1. This article proposes that the ABC Conjecture is true iff: c>0.

Multiplicative partitions of numbers with a large squarefree divisor

For each positive integer n, let f(n) denote the number of multiplicative partitions of n, meaning the number of ways of writing n as a product of integers larger than 1, where the order of the factors is not taken into account. It was shown by Oppenheim (J Lond Math Soc 1:205–211, 1926) that, as $$x\rightarrow \infty $$ , $$\begin{aligned} \max _{\begin{array}{c} n \le x \\ n\text { squarefree} \end{array}} f(n) = x/L(x)^{2+o(1)}, \end{aligned}$$ where $$L(x) = \exp \big (\log x\cdot \frac{\log \log \log {x}}{\log \log x}\big )$$ . Without the restriction to squarefree n, the maximum is the significantly larger quantity $$x/L(x)^{1+o(1)}$$ ; this was proved by Canfield et al. (J Number Theory 17:1–28, 1983). We prove the following theorem that interpolates between these two results: for each fixed $$\alpha \in [0,1]$$ , $$\begin{aligned} \max _{\begin{array}{c} n \le x\\ \mathrm {rad}(n) \ge n^{\alpha } \end{array}} f(n) = x/L(x)^{1+\alpha +o(1)}. \end{aligned}$$ We deduce, on the abc conjecture, a nontrivial upper bound on how often values of certain polynomials appear in the range of Euler’s $$\varphi $$ -function.

Arithmetic functions and the Cauchy product

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$).

The great Theorem of Pierre de Fermat as a special case abc conjecture and the proof of Andrew Beal hypothesis. Part 2.

The great Theorem of Pierre de Fermat as a special case abc conjecture and the proof of Andrew Beal hypothesis. Part 2.

Non-Wieferich primes under the abc conjecture

Assuming the abc conjecture, Silverman proved that, for any given positive integer a⩾2, there are ≫log⁡x primes p⩽x such that ap−1≢1(modp2). In this paper, we show that, for any given integers a⩾2 and k⩾2, there still are ≫log⁡x primes p⩽x satisfying ap−1≢1(modp2) and p≡1(modk), under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.

Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes

Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form an−bn, denoted P(an−bn), by essentially proving that for integers a>b>0, P(an−bn)≥n+1 for every n>2. Since then, the problem of finding bounds on the largest prime factor of Lehmer sequences, Lucas sequences or special cases thereof has been studied by many, most notably by Schinzel (1962), and Stewart (1975, 2013). In 2002, Murty and Wong proved, conditionally upon the abc conjecture, that P(an−bn)≫n2−ϵ for any ϵ>0. In this article, we improve this result for the specific case where b=1. Specifically, we obtain a more precise result, and one that is dependent on a condition we believe to be weaker than the abc conjecture. Our result actually concerns the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer a, P(Φn(a)), as we let n grow. We additionally prove some results related to the prime factorization of Φn(a). We also present a connection to Wieferich primes, as well as show that the finiteness of a particular subset of Wieferich primes is a sufficient condition for the infinitude of non-Wieferich primes. Finally, we use the technique used in the proof of the aforementioned results to show an improvement on average of estimates due to Erdős for certain sums.

A Crisis of Identification

Almost a decade after first publishing a purported proof of the abc conjecture, the prospects for Shinichi Mochizuki’s work remain unclear. Indeed, Mochizuki’s ideas are still met with puzzlement by many mathematicians. David Michael Roberts examines recent efforts by Peter Scholze and Jacob Stix to resolve the impasse and ponders the ultimate fate of Mochizuki’s proof.

Balancing non-Wieferich primes in arithmetic progressions

A prime is called a balancing non-Wieferich prime if it satisfies $$B_{p - \genfrac(){}{}{8}{p}} \not \equiv 0\pmod {p^{2}},$$ where $$\genfrac(){}{}{8}{p}$$ and $$B_n$$ denote the Jacobi symbol and the n-th balancing number respectively. For any positive integers $$k > 2$$ and $$n > 1$$ , there are $$\gg \log x / \log \log x$$ balancing non-Wieferich primes $$p \le x$$ such that $$p \equiv 1 \pmod {k}$$ under the assumption of the abc conjecture for the number field $$\mathbb {Q}(\sqrt{2})$$ (Proc. Japan Acad. Ser. A 92 (2016) 112–116). In this paper, for any fixed M, the lower bound $$\log x / \log \log x$$ is improved to $$(\log x/ \log \log x)(\log \log \log x)^{M}$$ .

The kernel of powerful numbers part II, with applications to the ABC conjecture

The kernel of powerful numbers part II, with applications to the ABC conjecture

The kernel function and applications to the ABC conjecture

The kernel function and applications to the ABC conjecture

Multiplicative dependence of the translations of algebraic numbers

In this paper, we first prove that given pairwise distinct algebraic numbers \alpha_1, \ldots, \alpha_n , the numbers \alpha_1+t, \ldots, \alpha_n+t are multiplicatively independent for all sufficiently large integers t . Then, for a pair (a,b) of distinct integers, we study how many pairs (a+t,b+t) are multiplicatively dependent when t runs through the set integers \mathbb Z . Assuming the ABC conjecture we show that there exists a constant C_1 such that for any pair (a,b)\in \mathbb Z^2 , a \ne b , there are at most C_1 values of t \in \mathbb Z such that (a+t,b+t) are multiplicatively dependent. For a pair (a,b) \in \mathbb Z^2 with difference b-a=30 we show that there are 13 values of t \in \mathbb Z for which the pair (a+t,b+t) is multiplicatively dependent. We further conjecture that 13 is the largest number of such translations for any such pair (a,b) and prove this for all pairs (a,b) with difference at most 10^{10} .