Building on work of Zhi-Hong Sun, we establish congruences for the special harmonic numbers $H_\lfloor p/9 \rfloor$ and $H_{\lfloor p/18 \rfloor}$ modulo $p$, which contain respectively three and four distinct arithmetic components. We also obtain a complete determination modulo $p$ of the corresponding families of sums of reciprocals of the type studied by Dilcher and Skula. Applications to the first case of Fermat's Last Theorem are considered.
Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$ with $x,y,z\in\mathbb Z$ if $(a,b,c)\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\lfloor T_x/a\rfloor+\lfloor T_y/b\rfloor+\lfloor T_z/c\rfloor$ with $x,y,z\in\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases…
How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes?
Institute of Mathematical Sciences, A CI of Homi Bhabha National Institute（马德拉斯数学科学研究所）；National Institute of Technology, Tiruchirappalli（国立特鲁奇拉帕利理工学院）
Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)=1\big\} $$ exists and equals $\frac{1}{ζ(k+1)}$, where $ζ$ is the Riemann zeta function.
This paper completes the classification of the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$, $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $α, β$ are negative. This result is equivalent to classifying all positive $α, β$ satisfying $ \lfloor{α\lceil{βx}\rceil}\rfloor - \lfloor{β\lceil{αx}\rceil}\rfloor \geq 0$ for all real $x$. The cl…
We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his original paper.
Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are infinitely many members of ${\mathbb P}^c$ having at most $R(c)$ prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and also when $c$ is large.
Let $π(n)$ denote the prime-counting function and let $$f(n)=\left|\left\lfloor\log n-\lfloor\log n\rfloor-0.1\right\rfloor\right|\left\lfloor\frac{\left\lfloor n/\lfloor\log n-1\rfloor\right\rfloor\lfloor\log n-1\rfloor}{n}\right\rfloor\text{.}$$ In this paper we prove that if $n$ is an integer $\ge 60184$ and $f(n)=0$, then $π(n)$ does not divide $n$. We also show that if $n\ge 60184$ and $π(n)$ divides $n$, then $f(n)=1$. In addition, we prove that if $n\ge 60184$ and $n/π(n)$ is an integer, then $n$ is a multiple of $\lfloor\log n-1\rfloor$ located in the interval $[e^{\lfloor\log n-1\rflo…
Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every $n\leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$. Let $π[n+g(n),n+f(n)+g(n)]$ denote the amount of prime num…
In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$ p(n) = \sum_{d|n} \sum_{k=1}^{d} \sum_{i_0 =1}^{\lfloor d/k \rfloor} \sum_{i_1 =i_0}^{\lfloor\frac{d- i_0}{k-1} \rfloor} \sum_{i_2 =i_1}^{\lfloor\frac{d- i_0 - i_1}{k-2} \rfloor} ... \sum_{i_{k-3}=i_{k-4}}^{\lfloor\frac{n- i_0 - i_1-i_2- ...-i_{k-4}}{3} \rfloor} \sum_{c|(d,i_0,i_1,i_2,...,i_{k-3})} μ(c) (\lfloor \frac{d-i_0-i_…
In this paper and its sequel we classify the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$ and $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. The relation $[f_α,f_β]\geq 0$ induces a preorder on the set of non-zero dilation factors $α, β$, which extends the divisibility partial order on positive integers. This paper treats the cases where at least one of the dilation parameters $α$ or $β$ is n…
Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor g_1 \rfloor = 3$, $\lfloor g_2 \rfloor = 13$, and $\lfloor g_3 \rfloor = 16381$. But with this $c$, $\lfloor g_4 \rfloor$ is a 4932-digit composite number. However, this slightly larger value of $c$, \[ c = 1.9287800 + 8.2843 \cdot 10^{-4933}, \] reproduces Wright's first three primes and generates a fourth: \[ \lfloor g_4 \rfloor = 19139664204631104984038373…
Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfl…
We show that the kernel $K(x,y)={1\over 2}+\lfloor {1\over xy}\rfloor -{1\over xy}$ ($0<x,y\leq 1$) has infinitely many positive eigenvalues and infinitely many negative eigenvalues. Our interest in this kernel is motivated by the appearance of the quadratic form $\sum_{m,n\leq N} K\bigl( {m\over N} , {n\over N}\bigr) μ(m)μ(n)$ in an identity involving the Mertens function.
The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$ satisfies a prime number theorem of the form $\sum_{n\le x} Λ(n)a(\lfloor n^c \rfloor) \sim C\, x$, and, furthermore, that it is deterministic for $c \in \mathbb R\setminus \mathbb Z$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor \bmod m$ has polynomial subword complexity.
We determine all pairs of real numbers $(α, β)$ such that the dilated floor functions $\lfloor αx\rfloor$ and $\lfloor βx\rfloor$ commute under composition, i.e., such that $\lfloor α\lfloor βx\rfloor\rfloor = \lfloor β\lfloor αx\rfloor\rfloor$ holds for all real $x$.
Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant $c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In particular, we prove the existence of the four infinite ex…
Mills proved that there exists a real constant $A>1$ such that for all $n\in \mathbb{N}$ the values $\lfloor A^{3^n}\rfloor$ are prime numbers. No explicit value of $A$ is known, but assuming the Riemann hypothesis one can choose $A= 1.3063778838\ldots .$ Here we give a first unconditional variant: $\lfloor A^{10^{10n}}\rfloor$ is prime, where $A=1.00536773279814724017\ldots$ can be computed to millions of digits. Similarly, $\lfloor A^{3^{13n}}\rfloor$ is prime, with $A=3.8249998073439146171615551375\ldots .$
Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based o…
The integral kernel $K(x,y) := \frac{1}{2} + \lfloor \frac{1}{xy} \rfloor - \frac{1}{xy}$ ($0<x,y\leq 1$) has connections with the Riemann zeta-function and a (recently observed) connection with the Mertens function. In this paper we begin a general study of the eigenfunctions of $K$. Our proofs utilise some classical real analysis (including Lebesgue's theory of integration) and elements of the established theory of square integrable symmetric integral kernels.
Let $λ$ denote the Liouville function. We show that the logarithmic mean of $λ(\lfloor α_1n\rfloor)λ(\lfloor α_2n\rfloor)$ is $0$ whenever $α_1,α_2$ are positive reals with $α_1/α_2$ irrational. We also show that for $k\geq 3$ the logarithmic mean of $λ(\lfloor α_1n\rfloor)\cdots λ(\lfloor α_kn\rfloor)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $α_i$. Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. The…
A generalized Beatty sequence is a sequence $V$ defined by $V(n)=p\lfloor{nα}\rfloor+qn +r$, for $n=1,2,\dots$, where $α$ is a real number, and $p,q,r$ are integers. These occur in several problems, as for instance in homomorphic embeddings of Sturmian languages in the integers. Our results are for the case that $α$ is the golden mean, but we show how some results generalise to arbitrary quadratic irrationals. We mainly consider the following question: For which sixtuples of integers $p,q,r,s,t,u$ are the two sequences $V=(p\lfloor{nα}\rfloor+qn +r)$ and $W=(s\lfloor{nα}\rfloor+tn +u)$ complem…
We show among others that the formula: $$ \lfloor n + \log_Φ\{\sqrt{5}(\log_Φ(\sqrt{5}n) + n) -5 + \frac{3}{n}\} - 2 \rfloor (n \geq 2), $$ (where $Φ$ denotes the golden ratio and $\lfloor \rfloor$ denotes the integer part) generates the non-Fibonacci numbers.
We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.
For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor αn+β\rfloor : n=1,2,3, \dots \rbrace$, where $α, β\in \mathbb{R}$ with $α>1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p)=\lfloor αn+β\rfloor.$ Next we obtain an asymptotic formula for number of primes $p$ of the form $p=\lfloor αn+β\rfloor$ which also satisfies $p \equiv f \pmod d$ where $f, d$ are integers with $1\leq…
We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(ψ(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor - n\log{n} - (2γ-1)n \Big\rvert $$ and $ ψ(n) $ - any positive function that increases unboundedly as $ n \to \infty $. The result is achieved under the hypothesis: $$ \Big \{\frac{n}{x} \Big \} \sim w_x $$ where $ w_x $ is uniformly distributed over $ [0,1) $ random variable with a values set $ \{0, \frac {1} {x}, \ldots, \frac{x-1}{x} \} $ and the value accepting…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfl…
Suppose that $α_1, α_2,β_1, β_2 \in\mathbb{R}$. Let $α_1, α_2 > 1$ be irrational and of finite type such that $1, α_1^{-1}, α_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range $1 < c < 12/11$. In this paper, it is proved that there exist infinitely many primes in the intersection of Beatty sequences $\mathcal{B}_{α_1,β_1} = \lfloorα_1 n + β_1\rfloor, \mathcal{B}_{α_2, β_2} = \lfloorα_2 n + β_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the inter…
Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ denotes the greatest integer function. We show how the error term in the asymptotic formula for $S_f(x)$ can be improved in some specific cases.
A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^α$ for some $α>0$. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. Furthermore, we extend Szemerédi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence $\{\lfloor{n^a}\rfloor\}_{n\in A}$ contains arbitrarily long a…
Given positive coprime integers $a$ and $b$ and a natural number $h$, we obtain reciprocity relations which can be used to quickly evaluate summations like $\sum_{i=1}^{h} \{\frac{ib}{a}\}^2$ and $\sum_{i=1}^{h} \lfloor \frac{ib}{a} \rfloor^2$, where $\lfloor x \rfloor$ and $\{x\}$ denote the floor function and the fractional part of $x$, respectively.
Let $α,β$ be real numbers such that $α>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.
Let $f(n)$ be an arithmetic function with $f(n) \ll n^α$ for some $α\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left( \left\lfloor \frac{x}{n_{1}n_{2}} \right\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.
We show that the slopes of the~$U_5$ operator acting on slopes of 5-adic overconvergent modular forms of weight~$k$ with primitive Dirichlet character~$χ$ of conductor~25 are given by either \[ \left\{{1/4}\cdot\lfloor\frac{8i}{5}\rfloor: i \in \N\right\}\text{or }\left\{{1/4}\cdot\lfloor\frac{8i+4}{5}\rfloor: i \in \N\right\}, \] depending on~$k$ and~$χ$.
We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.
We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\sqrt{n}$, the `large' exponents $\lfloor \frac{n}{3} \rfloor +1, \lfloor \frac{n}{2} \rfloor, n-1$, and for even $n \ge 4$, the additional value $\frac{n}{2}-1$. It is easy to see that no exponent in $[\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\lfloor \frac{n}{3}\rfloor +2,\frac{n}{2}-2]$ is possible. Extend…
We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.
Let $p$ be a prime number and let $k\geq 2$ be an integer. In this article we study the semi-simple reductions modulo $p$ of two-dimensional irreducible crystalline $p$-adic Galois representations with Hodge-Tate weights $0$ and $k-1$ and large slopes. Berger--Li--Zhu proved by using the theory of $(φ,Γ)$-modules that this reduction is constant when the slope is larger than $\lfloor\frac{k-2}{p-1}\rfloor$. Recently, Bergdall--Levin improved this bound to $\lfloor\frac{k-1}{p}\rfloor$ by using the theory of Kisin modules. In this article, under the extra assumptions $p>3$ and $p+1\nmid k-1$, we…
We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} φ\left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $φ$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers.
Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give t…
We study the distribution of squares in a Piatetski-Shapiro sequence $\left(\lfloor n^c\rfloor\right)_{n\in\mathbb N}$ with $c>1$ and $c\not\in\mathbb N$. We also study more general equations $\lfloor{n^c}\rfloor = sm^2$, $n,m\in \mathbb N$, $1\le n \le N$ for an integer $s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.
We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell(m,x)={\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor+b_i}_i}=m{\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor}_i}$$ where $m=\prod^w_{i=1}p^{b_i}_i$. We prove certain interesting properties of $\ell(m,x)$ and derive a few other results and congruences.
The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} β^{-w(m)}\), where \(β\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce criteria for the algebraic independence of such values. Our criteria are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{…
We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor jβ\rfloor $ against $j$ for every quadratic unit $β\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor β(j+G_i)\bigr\rfloor = \lfloor βj\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $β^i$.
