Journal of Number Theory最新文献

The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in $N$ that does not contain $x_1,x_2,y_1,y_2$ with $x_1+x_2=y_1+y_2$. Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form $x_1+x_2$ to arbitrary linear forms $c_1{x}_1+\dots +c_h{x}_h$; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when $c_i=n^{i-1}$ for some $n\ge 2$, and also when $h=2,c_1=2,c_2\ge 4$, the “structured” domain. We also contrast the “enigmatic” domain when $h=2,c_1=2,c_2=3$ with the “structured” domain, and give upper bounds on the growth rates in both cases.

We prove explicit lower bounds for linear forms in two p-adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, $|\Lambda {|}_p=|{\alpha }_1^{b_1}-{\alpha }_2^{b_2}{|}_p$ (and corresponding explicit upper bounds for $v_p\left(\Lambda \right)$), where ${\alpha }_1,{\alpha }_2$ are numbers that are algebraic over $Q$ and $b_1,b_2$ are positive rational integers.

This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for $v_p\left(\Lambda \right)$ has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on $b_1$ and $b_2$) is $\log B$, instead of ${(\log B)}^2$ as in the work of Bugeaud and Laurent in 1996.

For squarefree $d>1$, let M denote the ring class field for the order $Z\left[\sqrt{-3d}\right]$ in $F=Q\left(\sqrt{-3d}\right)$. Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of $Q$ such that E and F have the same discriminant. Define the real cube roots $v={(a+b\sqrt{d})}^{1/3}$ and $v^{\prime }={(a-b\sqrt{d})}^{1/3}$, where $a+b\sqrt{d}$ is the fundamental unit in $Q\left(\sqrt{d}\right)$. We prove that E can be taken as $Q(v+v^{\prime })$ if and only if $v\in M$. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if $v\in M$, and we also show that the norm of the relative discriminant of $F\left(v\right)/F$ lies in $\{1,3^6\}$ or $\{3^8,3^{18}\}$ according as $v\in M$ or $v\notin M$. We then prove that v is always in the ring class field for the order $Z\left[\sqrt{-27d}\right]$ in F. Some of the results above are extended for subsets of $Q\left(\sqrt{d}\right)$ properly containing the fundamental units $a+b\sqrt{d}$.

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip.

Let E and $E^{\prime }$ be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if $E\left(F_p\right)\simeq E^{\prime }\left(F_p\right)$ but $E\left(F_{p^2}\right)\not\simeq E^{\prime }\left(F_{p^2}\right)$. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.

This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number $C_3\left(X\right)$ of Carmichael numbers up to X with three prime factors is $\le \sqrt{X}$ for all $X\ge 1$. He showed that his conjecture is true for $X\le {10}^{16}$ and $X>{10}^{126}$. In this article, it is shown that the conjecture is true for $X\le {10}^{24}$ and $X>2⁎{10}^{40}$. In both cases, analytical methods establish the conjecture for large X and tables of Carmichael numbers are used for small X.

Let $U$ be the set of positive odd numbers that can not be written in the form $p+2^k$. Recently, by analyzing possible prime divisors of b, Chen proved $b\ge 11184810$ and $\omega \left(b\right)\ge 7$ if an arithmetic progression $a\left(\mathrm{mod}b\right)$ is in $U$, with $\omega \left(b\right)=7$ if and only if $b=11184810$, where $\omega \left(n\right)$ is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove $b\ge 11184810$ and provide all possible values of a if $a\left(\mathrm{mod}11184810\right)$ is in $U$. Moreover, we explicitly construct nontrivial arithmetic progressions $a\left(\mathrm{mod}b\right)$ in $U$ with $\omega \left(b\right)=8$, 9, 10, or 11, and provide potential nontrivial arithmetic progressions $a\left(\mathrm{mod}b\right)$ in $U$ such that $\omega \left(b\right)=s$ for any fixed $s\ge 12$. Furthermore, we improve the upper bound estimate of numbers of the form $p+2^k$ by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 10¹⁶. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to $616,000$ by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity s in any global function field of genus g over $F_q$ for $q\ge (1.14+0.16g)e^{6.5+0.97g}{s}^2$. In particular, there is no covering system of $F_q\left[x\right]$ with distinct moduli for $q\ge 759$.

We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve $E/Q$ by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to $E/Q$ extended to chosen families of cyclic extensions of fixed degree.