T. N. Venkataramana
We prove that the gcd of certain infinite number of integers associated to generalised arithmetic progressions remains bounded independent of the progression. Using this we also get bounds on the indices of certain congruence subgroups of arithmetic groups.
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We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.
In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
A group of infinite products over low-order rational polynomials evaluated at the sequence of prime numbers is loosely called the Hardy-Littlewood constants. In this manuscript we look at them as factors embedded in a super-product over primes, semiprimes, 3-almost primes etc. Numerical tables are derived by transformation into series over k-almost prime zeta functions. Alternative product representations in a basis of k-almost prime products associated with Euler's formula for the Riemann zeta function are also pointed out.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
本文证明对任意整数N≥2和m≥1，存在无穷多对实二次域，其类数均被N整除，且当N的奇部大于1时，类群含阶为该奇部的元素。
Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes
本文证明在模无穷大素数整数环中，Montgomery 归约因子 $R'$ 的像在有理数域上具有超越性，其中 $R'$ 定义为满足特定不等式的 $2$ 的幂的模逆。
Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations, then $A(x) \geq (\log x)/\log h)-w_0$, where $A(x)$ counts the number of integers $a \in A$ such that $1 \leq a \leq x$.
We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd integers.
There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.
Construction of diagonal quintic threefolds with infinitely many rational points
本研究构造了一族定义在有理数域上、各含无穷多有理点的对角五次三维簇，并证明存在无穷多组互素整数四元组，使得对应五次方程对合适的依赖参数有无穷多正整数解。
Some results on naive transcendence in the ring of integers modulo infinitely large primes
本文在无限大素数模整环中研究超越性问题，改进了Anzawa-Funakura和Luca-Zudilin的结果，提出新的朴素超越数例子，并在ABC猜想下证明了某些数的无理性。
Pythagoras numbers for infinite algebraic fields
研究了无限代数数域的毕达哥拉斯数，证明某些数域的毕达哥拉斯数为无限，同时构造了毕达哥拉斯数为有限值的数域。
Problems on infinite sumset configurations in the integers and beyond
本文围绕正密度自然数集中的无限和集及其他无限构型，梳理相关最新进展，并系统探讨待解决问题与存在的障碍。
Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.
Cardinalities of Jordan Diophantine sets of upper triangular integer matrices
该研究确定了上三角整数矩阵环中Jordan D(N)-集存在无限集的N的条件，明确了有限集的最大基数，还给出了两类特殊无限集存在的充要条件。
Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $Γ$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $Γ$ is boundedly generated as an abstract group.
Every arithmetic progression contains infinitely many $b$-Niven numbers
该研究针对数论中$b$-Niven数的分布问题，证明了任意等差数列内都存在无穷多个$b$-Niven数，丰富了Niven数的分布理论成果。
Inspired by the Erdös-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem and Sźmeredis Theorem that the primes and all subsets of positive upper density are AP-sets. We prove that (1, 1,..., 1) is a solution to the equation Mx = 0 where M is an integer matrix whose null space has dimension at least 2, if and only if the equation has infinitely many solutions such that the coordinates of each solution are elements in the same AP. Thi…
Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes
本文探讨无限大素数模整数环的代数与模型论性质，结合非标准分析方法扩展超越数理论中的判别准则。
In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$ practical. We also prove that there are infinitely many practical numbers of the form $q^4+2$ with $q$ practical, and that there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=6$ (or $\gcd(a,b,c)=4$).
Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $(δ_1,\ldots,δ_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{δ_j}.$$
In 1960, W. Sierpinski proved that there are infinitely many positive odd numbers $k$, such that for any positive integer $n$, $k\times2^n+1$ is a composite number. Such numbers are called "Sierpinski numbers". In this study, by using covering systems and the theory of cyclotomic polynomials, the following theorem is proved: for any integer $m>1$, there are infinitely many integers $k$ satisfying $k\not\equiv-1\!\pmod{q}$ for any prime number $q|(m-1)$, such that for any positive integer $n$, $km^n+1$ is a composite number. These positive integers $k$ are called "Sierpinski numbers based $m$".…
We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime $q$ and any positive rational integer $m$, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by $q^{m}$. (…
We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they converge to unity irrespective of which positive integer is chosen initially.
Let $ζ_k$ be a $k$-th primitive root of unity, $m\geqϕ(k)+1$ an integer and $Φ_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+ζ_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|Φ_k(m)|\}$. Moreover we also discuss whether the two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.
The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-β}$ of numbers whose representation uses only non-negative powers of $-β$, the so-called $(-β)$-integers. We describe the distances between consecutive elements of $\Z_{-β}$. In case that this set is non-trivial we associate to $β$ an infinite word $\boldsymbol{v}_{-β}$ over an (in general infinite) alphabet. The self-similarity of $\Z_{-β}$, i.e., the property $-β\Z_{-β}\subset \Z_{-β}$, allows us to find a morphism under…
A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we…
Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $σ$, there exists a subset $A$ of $\mathbb{Z}^{2d}$ such that $A$ and $σ(A)$ form a co-minimal pair. The …
Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$, then the Hilbert class field of $\K$ is $\H_\K=\K(\sqrt{2})$. Finally, we find a criterion to decide if an ideal of the ring of integers or $\K$ is principal or non-principal.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.
For an algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$, an important subgroup of the ideal class group $Cl_{K}$ is the {\it Pólya group}, denoted by $Po(K)$, which measures the failure of the $\mathcal{O}_{K}$-module $Int(\mathcal{O}_{K})$ of integer-valued polynomials on $\mathcal{O}_{K}$ from admitting a regular basis. In this paper, we prove that for any integer $n \geq 2$, there are infinitely many totally real bi-quadratic fields $K$ with $|Po(K)| = 2^{n}$. In fact, we explicitly construct such an infinite family of number fields. This extends an infinite family of bi-q…
A set of integers $S \subset \mathbb{N}$ is an $α$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $α$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^α,y^α,z^α,w^α\}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $α$-strong infinite Sidon sets when $0 \leq α< 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $α$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density …
By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $\mathcal{S}_n:=\{s_1, ..., s_n\}$ and $H_f(\mathcal{S}_n):=\sum_{k=1}^{n}\frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $\mathcal{S}$ of positive integers, $H_f(\mathcal{…
Expressing an integer as a sum of cubes of polynomials
本文证明无穷多整数可表示为四个整系数多项式立方和，且每个整数可表示为五个多项式立方和，并给出1和2的具体恒等式。
An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many positive integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
Motivated by a WhattsApp message, we find out the integers $x> y\ge 1$ such that $(x+1)/(y+1)=(x\circ(y+1))/(y\circ (x+1))$, where $\circ$ means the concatenation of the strings of two natural numbers (for instance $783\circ 56=78356$). The discussion involves the equation $x(x+1)=10y(y+1)$, a slight variation of Pell's equation related to the arithmetic of the Dedekind ring $\mathbb{Z}[\sqrt{10}]$. We obtain the infinite sequence $\mathcal{S}=\{(x_n,y_n)\}_{n\ge 1}$ of all the solutions of the equation $x(x+1)=10y(y+1)$, which tourn out to have limit $1/\sqrt{10}$. The solutions of the initia…
Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity …
Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results of Liu and Fang in 2016 and obtain some results on additive complements. For example, we prove that there exist additive complements $A$ and $B$ such that $\limsup_{x\to+\infty} A(x)B(x)/x= 2$ and $A(x)B(x) - x = 1$ for infinitely positive integers $x$.
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。
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.
We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When $(ra,sb)=1$, we show that $N \le 3$ except for a finite number of cases all of which satisfy $\max(a,b,r,s, x,y) < 2 \cdot 10^{15}$ for each solution; when $(a,b)>1$, we show that $N \le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of infinite families of cases giving N=3 solutions.
On the representation of integer as sum of a square-free number and a prime of special type
本文证明存在无穷多个整数可表示为一个无平方因子整数与一个满足特定丢番图不等式条件的素数之和，其中系数α为无理数。
We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely hard to solve. In this paper we represent our attempts to explore the field by means of experimental mathematics. Having computed the values of ||n|| up to 10^12 we present our observations. One of them (if true) implies that there is an infinite num…
We present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.
Dynamics of the Fibonacci Order of Appearance Map
本文研究Fibonacci出现阶映射的迭代动力学，证明存在无穷多整数恰经k步到达不动点，构造趋向12·5^k型不动点的无穷族，并另证所有正整数有限步内到达不动点。
We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if \begin{equation*} \frac{(q^{r-tk}, q^{mk-(r-tk)}; q^{mk})_\infty}{(q^r,q^{mk-r}; q^{mk})_\infty} =: \sum_{n=0}^\infty c_nq^n, \end{equation*} for certain integers $k$, $m$ $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretatio…
Some Results on Zumkeller Numbers
本文研究了Zumkeller数的性质，证明了存在任意长度的连续Zumkeller数段，大于94185的整数均可表示为两Zumkeller数之和，并探讨了其与practical数、平方数及素数的加法表示关系。
We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.
Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.
Generalized Sierpiński and Riesel numbers of the form $tb^t+α$
本文证明对任意非零整数α，形如tb^t+α的数中存在无穷多个b-Sierpiński数和b-Riesel数，并在b+1非2的幂时存在无穷多个b-Brier数。
Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $π\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $π\circ F=f\circπ$. In this paper we relate the finite and infinite noti…
A new characterization for the Lucas-Carmichael Integers and sums of base-$p$ digits
本文通过基-$p$ 位数和给出 Lucas-Carmichael 整数的充分必要刻画，研究其相关性质，并在素数 $k$-元组猜想下证明其无穷性。
Universal quadratic forms and Northcott property of infinite number fields
本文证明无限次全实扩张上泛二次型的存在性与全正整数集合的 Northcott 性质不相容，并据此推出在固定素数次全实 Galois 域的合成域及次数为 3d 的合成域上不存在经典泛二次型。
We investigate the number $R_3(n)$ of representations of $n$ as the sum plus the product of three positive integers. On average, $R_3(n)$ is $\frac{1}{2}\log^2 n$. We give an upper bound for $R_3(n)$ and an upper bound for the number of $n \leq N$ such that $R_3(n) = 0$. We conjecture that $R_3(n)= 0$ infinitely often.
We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each $k\in\{12,16,18,20,22,26\}$, we give explicit rational primes $ł$ such that the fixed field of the mod-$ł$ representation attached to the unique normalized cusp eigenforms of weight $k$ on $\Sl_2(\Z)$ has an infinite class field tower. Under a conjecture of Hardy and Littlewood, we further prove that there exist infinitely many such primes for each $k$ (in the above list). Second, given a non-CM curve $E/\Q$, we show that t…
In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of an integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an e…
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