Hongjian Li

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20222026

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math.NT2026

Critical Pairs for Mixed Restricted Sumsets in Prime Fields

Weilin Zhang, Hongjian Li

Let $p$ be an odd prime and let $A, B \subseteq F_{p}$ be nonempty and distinct, with $∣ A ∣ \geq ∣ B ∣$ . We give a complete classification of the pairs for which the mixed restricte…

math.NT2026

Reducibility of $r x^{m} + p^{e} f (x)$ for Large Primes $p$

Weilin Zhang, Hongjian Li

We study the reducibility over $Q$ of $F_{p, e} (x) = r x^{m} + p^{e} f (x)$ , where $r \in Z ∖ {0}$ , $f \in Z [x]$ , and $0 \leq m < n := deg f$ , for primes $p…

math.NT2026

The Fermat-Type Matrix Equation over $GL_{2} (Z)$

Hongjian Li, Weilin Zhang

We determine the complete ordered solution set of the Fermat-type matrix equation $X^{n} + Y^{n} = Z^{n}$ in $GL_{2} (Z)$ for every integer $n \geq 3$ , expressed in terms of simu…

math.NT2026

The Matrix Pythagorean Equation over $GL_{2} (Z)$

Hongjian Li, Weilin Zhang

In this paper, we study the ordered solutions of the matrix Pythagorean equation $X^{2} + Y^{2} = Z^{2}$ over $GL_{2} (Z)$ . Exploiting the independent sign symmetry of the eq…

math.NT2026

A Multiplicative Fourier Proof of the Length-Four Index Conjecture

Hongjian Li, Pingzhi Yuan, Shijie Yuan +1

Let $C_{n}$ be a cyclic group of order $n$ . We prove that if $(n, 6) = 1$ , then every minimal zero-sum sequence of length four over $C_{n}$ has index one, thereby resolving the length-fou…

math.NT2025

Fermat's and Catalan's equations over $M_{2} (Z)$

Hongjian Li, Pingzhi Yuan

Let $A = (a c b d) \in M_{2} (Z)$ be a given matrix such that $b c \neq = 0$ and let $C (A) = {B \in M_{2} (Z) : A B = B A}$ . In this pap…

math.NT2025

Faithful Decomposition of Rationals

Sunben Chiu, Pingzhi Yuan, Hongjian Li

If an irreducible fraction $\frac{m}{n} > 0$ can be decomposed into the sum of several irreducible proper fractions with different denominators, and the positive number smaller than $\f…

math.NT2025

Some permutation polynomials via linear translators

Xuan Pang, Pingzhi Yuan, Hongjian Li

Permutation polynomials with explicit constructions over finite fields have long been a topic of great interest in number theory. In recent years, by applying linear translators of…

math.NT2025

On the Elementary Symmetric Functions of ${1, 1/2, \dots, 1/ n} \ {1/ i}$

Weilin Zhang, Hongjian Li, Sunben Chiu +1

In 1946, P. Erdős and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1, 1/2$ , $\cdots, 1 /…

math.NT2025

On the representation of rational numbers via Euler's totient function

Weilin Zhang, Fengyuan Chen, Hongjian Li +1

Let $b > 1$ be an odd positive integer and $k, l \in N$ . In this paper, we show that every positive rational number can be written as $φ (m^{2}) / (φ (n^{2}))^{b}$ and $φ(k(m^{2…

math.NT2025

The order of appearance of the product of the first and second Lucas numbers

Hongjian Li, Huiming Xiao, Pingzhi Yuan

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $(U_{n})_{n \geq 0}$ and the second Lucas sequence $(V_{n})_{n \geq 0}$ are defined respec…

math.NT2022

The matrix equation $a X^{m} + b Y^{n} = c I$ over $M_{2} (Z)$

Hongjian Li, Pingzhi Yuan

Let $N$ be the set of all positive integers and let $a, b, c$ be nonzero integers such that $g cd (a, b, c) = 1$ . In this paper, we prove the following thr…

math.NT2022

Positive rational number of the form $φ (k m^{a}) / φ (l n^{b})$

Hongjian Li, Pingzhi Yuan, Hairong Bai

Let $k, l, a$ and $b$ be positive integers with $max {a, b} \geq 2$ . In this paper, we show that every positive rational number can be written as the form $φ (k m^{a}) / φ (l n^{b})$ ,…

Critical Pairs for Mixed Restricted Sumsets in Prime Fields

Reducibility of rxm+pef(x) for Large Primes p

The Fermat-Type Matrix Equation over GL2​(Z)

The Matrix Pythagorean Equation over GL2​(Z)

A Multiplicative Fourier Proof of the Length-Four Index Conjecture

Fermat's and Catalan's equations over M2​(Z)

Faithful Decomposition of Rationals

Some permutation polynomials via linear translators

On the Elementary Symmetric Functions of {1,1/2,…,1/n}\{1/i}

On the representation of rational numbers via Euler's totient function

The order of appearance of the product of the first and second Lucas numbers

The matrix equation aXm+bYn=cI over M2​(Z)

Positive rational number of the form φ(kma)/φ(lnb)

Reducibility of $r x^{m} + p^{e} f (x)$ for Large Primes $p$

The Fermat-Type Matrix Equation over $GL_{2} (Z)$

The Matrix Pythagorean Equation over $GL_{2} (Z)$

Fermat's and Catalan's equations over $M_{2} (Z)$

On the Elementary Symmetric Functions of ${1, 1/2, \dots, 1/ n} \ {1/ i}$

The matrix equation $a X^{m} + b Y^{n} = c I$ over $M_{2} (Z)$

Positive rational number of the form $φ (k m^{a}) / φ (l n^{b})$