8 citations · 13 across the 7 of their papers we have counts for
7 papers · 1 filter
Cyclotomic Congruences and Lucas Sequences
Tyler Ross, Zhongyan Shen, Tianxin Cai
In this paper, we extend the -adic valuations originally obtained by Carmichael for the sequences obtained by applying Möbius inversion to Lucas sequences to -adic congruence…
The -adic Valuations of Möbius Duals of Lucas Sequences
Tyler Ross, Zhongyan Shen, Tianxin Cai
In this paper, we extend the -adic valuations of the Möbius duals of Lucas sequences, originally obtained by Carmichael for regular Lucas sequences to irregular Lucas sequences.…
Arithmetic properties of polynomials
Yong Zhang, Zhongyan Shen
In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for with nonzero integers $a\eq…
Super congruences involving alternating harmonic sums modulo prime powers
Zhongyan Shen, Tianxin Cai
In 2014, Wang and Cai established the following harmonic congruence for any odd prime and positive integer , \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathca…
A congruence involving alternating harmonic sums modulo
Zhongyan Shen, Tianxin Cai
In 2014, Wang and Cai established the following harmonic congruence for any odd prime and positive integer , \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathca…
A congruence involving harmonic sums modulo
Tianxin Cai, Zhongyan Shen, Lirui Jia
In 2014, Wang and Cai established the following harmonic congruence for any odd prime and positive integer , \begin{equation*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}…
Figurate primes and Hilbert's 8th problem
Tianxin Cai, Yong Zhang, Zhongyan Shen
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…