LLL & ABC

ABC implies primitive prime divisors in arithmetic dynamic

Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that p | f^n(z) and p does not divide f^m(z) for all positive integers m < n. We prove the same result unconditionally for function fields of characteristic 0 when f is not isotrivial.

abc triples

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…

ABC猜想：$ABC = 2^m p^n q^r$与费马或梅森素数

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。

$n^2+1$ 的最大素因子与亚指数 $ABC$ 的改进

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。

关于素数分解中加权平均重数的一个注记

A remark on weighted average multiplicities in prime factorisation

本文推广ABC三元组的质量定义，引入加权平均重数（WAM），证明其实部小于1时无界、大于1时全纯有界，并揭示其与最大素因子重数及ABC猜想的联系。

Differences of Harmonic Numbers and the $abc$-Conjecture

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 …

ABC implies a Zsigmondy principle for ramification

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $β\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.

ABC 推出拉马努金的 tau 函数错过几乎所有素数

ABC implies that Ramanujan's tau function misses almost all primes

在 ABC 猜想假设下，证明了 tau 函数的素数值密度为 O(X^{13/22})，表明 tau 函数错过大部分素数，同时提出 S(X) 可能无限且有预测阶数。

abc猜想蕴含数域中固定基的无穷多个非Wieferich位

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。

Smooth solutions to the abc equation: the xyz Conjecture

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…

A note on $p$-rational fields and the abc-conjecture

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.

关于形如 $(1,c-1,c)$ 的 $abc$ 三元组

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。

Large Tate--Shafarevich orders from good $abc$ triples

Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The la…

Insulators of ABC Solutions

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.

Determining the primes of bad reduction of CM curves of genus 3

In this paper we introduce a new problem called the Isogenous Embedding Problem (IEP). The existence of solutions to this problem is related to the primes of bad reduction of CM curves of genus $3$ and we can detect potentially good reduction in absence of solutions. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. We were able to prove the reduction type of some particular curves at certain primes that were open cases in [LLLR21].

Arithmetic properties of blocks of consecutive integers

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.

A note on balancing sequences and application to cryptography

In this paper, we prove the lower bound for the number of balancing non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$ such that $p\equiv\pm1 \pmod{r}$, under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{2})$. Further, we discuss some applications of balancing sequences in cryptography.

Sums of two cubes as twisted perfect powers, revisited

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^αC^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime exponents $p$, no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain {\it symplectic criteria}, we are able to make some conditional statements about still more values of $q$, a sample such result is that, for all but $O(\sqrt{x}/\log x)$ primes $q$ up to $x$, the equa…

算术级数中几乎完美的非齐次幂

Almost perfect inhomogeneous powers in arithmetic progression

本文研究有限素数集S定义的整数集H_S中算术级数的长度，证明abc猜想可推出其项数有界，并推广到其他集合。

多边形数序列中的完美幂

Perfect powers in sequences of polygonal numbers

研究多边形数序列中完美幂的丢番图方程，利用模方法、超几何方法、对数线性形式和大量计算，扩展了已知解的范围。

通过秩3格和第二向量进行因数分解和幂除数问题的研究

On Factoring and Power Divisor Problems via Rank-3 Lattices and the Second Vector

本文提出基于秩3格和第二向量的算法，改进了因数分解、幂和差分解及r-幂除数问题的复杂度。