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Tianxin Cai

Publications and source records attributed to Tianxin Cai.

At least 19 recordsLinked to original sources
arXiv2026

Cyclotomic Congruences and Lucas Sequences

In this paper, we extend the $p$-adic valuations originally obtained by Carmichael for the sequences obtained by applying Möbius inversion to Lucas sequences to $p$-adic congruences, from which we immediately derive corresponding congruences for Lucas sequences. As a corollary, we also establish some constraints on the entry point behavior of primes in Lucas sequences, on the basis of which we conjecture the presence of a strong Chebyshev-like bias in real regular Lucas sequences.

math.NT↗
arXiv2025

The $p$-adic Valuations of Möbius Duals of Lucas Sequences

In this paper, we extend the $p$-adic valuations of the Möbius duals of Lucas sequences, originally obtained by Carmichael for regular Lucas sequences to irregular Lucas sequences. We conclude with a brief observation about the relationship of these valuations to the existence of Wall-Sun-Sun primes.

math.NT↗
arXiv2019

A Generalization of A Result of Gauss on Primitive Root

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.

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arXiv2019

On the divisor problem with congruence conditions

Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $Δ(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\geq (q_1q_2)^{1+\varepsilon}, 1\leq r_i\leq q_i$, and $(r_i, q_i)=1$ ($i=1, 2$). We study the power moments and sign changes of $Δ(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $Δ(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm Δ(q_1q_2x; r_1, q_1, r_2, q_2)> c_5x^\frac{1}{4}$ always holds.

math.NT↗
arXiv2017

Perfect numbers and Fibonacci primes (III)

In this article, we consider the Diophantine equation $σ_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.

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arXiv2016

On the Lucas Property of Linear Recurrent Sequences

We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with $0 \leq n_{i} \leq p-1,n,n_{i}\in\mathbb{N}$. In this note, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.

math.NT↗
arXiv2016

A congruence involving harmonic sums modulo $p^αq^β$

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $ Z(n)=\sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{1}{ijk}$ and $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $α,~β$, \begin{equation*} Z(p^αq^β)\equiv 2(2-q)(1-\frac{1}{q^{3}})p^{α-1}q^{β-1}B_{p-3}\pmod{p^α} \end{equation*} and the necessary and sufficient condition for \begin{equation*} Z(p^αq^β)\equiv 0\pmod{p^αq^β}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^α||n$, $α\geq1$, \begin{eqnarray} \nonumber Z(n)\equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{2n}{p})B_{p-3}\pmod{p^α}. \end{eqnarray}

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arXiv2016

On the Sign Changes of a Weighted Divisor Problem

Let $S\big(x; \frac{a_1}{q_1}, \frac{a_2}{q_2}\big)=\mathop{{\sum}'}_{mn\leq x} \cos\big(2πm\frac{a_1}{q_1}\big)\sin\big(2πn\frac{a_2}{q_2}\big)$ with $x\geq q_1q_2, 1\leq a_i\leq q_i$, and $(a_i, q_i)=1$ ($i=1, 2$). We study the sign changes of $S\big(x; \frac{a_1}{q_1}, \frac{a_2}{q_2}\big)$, and prove that for a sufficiently large constant $C$, $S\big(x; \frac{a_1}{q_1}, \frac{a_2}{q_2}\big)$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm S\big(t; \frac{a_1}{q_1}, \frac{a_2}{q_2}\big)> c_5 (q_1q_2)^\frac{3}{4}t^\frac{1}{4}$ always holds.

math.NT↗
arXiv2015

A congruence involving alternating harmonic sums modulo $p^αq^β$

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain the congruences for distinct odd primes $p,~q$ and positive integers $α,~β$, \begin{equation*} \sum\limits_{i+j+k=p^αq^β\atop{i,j,k\in\mathcal{P}_{pq}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk}\equiv\frac{7}{8}(2-q)(1-\frac{1}{q^{3}})p^{α-1}q^{β-1}B_{p-3}\pmod{p^α} \end{equation*} and \begin{equation*} \sum\limits_{i+j+k=p^αq^β\atop{i,j,k\in \mathcal{P}_{pq}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}(q-2)(1-\frac{1}{q^{3}})p^{α-1}q^{β-1}B_{p-3}\pmod{p^α}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^α||n$, $α\geq1$, \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{(-1)^{i}}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})\frac{n}{2p}B_{p-3}\pmod{p^α} \end{eqnarray} and \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{7n}{8p})B_{p-3}\pmod{p^α}. \end{eqnarray}

math.NT↗
arXiv2015

Super congruences involving alternating harmonic sums modulo prime powers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} (\bmod p^{r}), \end{equation*} where $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we establish a combinational congruence of alternating harmonic sums for any odd prime $p$ and positive integers $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}p^{r-1}B_{p-3} (\bmod p^{r}). \end{equation*} For any odd prime $p\geq 5$ and positive integers $r$, we have \begin{align} &4\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}}+3\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}} \nonumber\\&\equiv\begin{cases} \frac{216}{5}pB_{p-5}\pmod{p^{2}}, if r=1, \\ \frac{36}{5}p^{r}B_{p-5}\pmod{p^{r+1}}, if r>1. \\ \end{cases}\nonumber \end{align} For any odd prime $p> 5$ and positive integers $r$, we have \begin{align} &\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}}}{i_{1}i_{2}i_{3}i_{4}i_{5}}+2\sum\limits_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}=2p^{r}\atop{i_{1}, i_{2}, i_{3}, i_{4}, i_{5}\in \mathcal{P}_{p}}}\frac{(-1)^{i_{1}+i_{2}}}{i_{1}i_{2}i_{3}i_{4}i_{5}} \nonumber\\&\equiv\begin{cases} 12B_{p-5}\pmod{p}, if r=1,\\ 6p^{r-1}B_{p-5}\pmod{p^{r}}, if r>1. \end{cases}\nonumber \end{align}

math.NT↗
arXiv2014

Figurate primes and Hilbert's 8th problem

In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with figurate primes and Hilbert's 8th problem, including Goldbach's conjecture, twin primes conjecture and Catalan's conjecture as well.

math.NT↗
arXiv2014

Perfect Numbers and Fibonacci Primes (II)

In this paper, we study the diophantine equation ${{σ}_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are $n={{F}_{2k+1}}{{F}_{2k+2m+1}}$, where both ${{F}_{2k+1}}$ and ${{F}_{2k+2m+1}}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${{σ}_{2}}(n)-{{n}^{2}}=2n+5$ has infinitely many solutions.

math.NT↗
arXiv2014

A New Generalization of Fermat's Last Theorem

In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where $A,B,C,D,n \in\ZZ_{+}$ and $n\geq3$, which may be regarded as a generalization of Fermat's equation $x^n+y^n=z^n$. When $\gcd(A,B,C)=1$, $(1)$ is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for $\gcd(A,B,C)=p^k$ where $p$ is an odd prime. In particular, for $k=1$ we prove that $(1)$ has no nonzero integer solutions when $n=3$ and we conjecture that it is also true for any prime $n>3$. Finally, we consider equation $(1)$ in quadratic fields $\mathbb{Q}(\sqrt{t})$ for $n=3$.

math.NT↗
arXiv2014

Perfect numbers and Fibonacci primes

In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.

math.NT↗
arXiv2013

A variety of Euler's conjecture

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions $n,b,a_i\in\mathbb{Z}^+,i=1,2,\ldots,s-1,s\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\in\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\in\mathbb{Z}^+$ and for $s\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\in\mathbb{Z}[t_1,t_2,\ldots,t_{s-3}]$ with positive value satisfying this Diophantine system.

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