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Hao Pan

Publications and source records attributed to Hao Pan.

At least 91 records · Page 5Linked to original sources
arXiv2019

Consecutive primes and Legendre symbols

Let $m$ be any positive integer and let $δ_1,δ_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=δ_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=δ_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any distinct $i$ and $j$ among $0,1,\ldots,m$.

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arXiv2019

A $q$-analogue of Wilson's congruence

Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{σ\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,σ} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\mathfrak f}_{n-1}(q)\equivμ(n)\pmod{Φ_n(q)}, $$ where $μ$ denotes the Möbius function and $Φ_n(q)$ is the $n$-th cyclotomic polynomial.

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arXiv2019

The Green-Tao theorem for Piatetski-Shapiro primes

Let $m\geq 3$. Suppose that $$ 1-2^{-2^{m^24^m}}<γ<1. $$ Then the set $$ \{p\text{ prime}:\, p=[n^{\frac1γ}]\text{ for some }n\in{\mathbb N}\} $$ contains infinitely many non-trivial $m$-term arithmetic progressions.

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arXiv2018

Some permutations over ${\mathbb F}_p$ concerning primitive roots

Let $p$ be an odd prime and let ${\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\mathbb F}_p$. Define the permutation $τ_g:\,{\mathcal H}_p\to{\mathcal H}_p$ by $$ τ_g(b):=\begin{cases} g^b,&\text{if }g^b\in{\mathcal H}_p,\\ -g^b,&\text{if }g^b\not\in{\mathcal H}_p,\\ \end{cases} $$ for each $b\in{\mathcal H}_p$, where ${\mathcal H}_p=\{1,2,\ldots,(p-1)/2\}$ is viewed as a subset of ${\mathbb F}_p$. In this paper, we investigate the sign of $τ_g$. For example, if $p\equiv 5\pmod{8}$, then $$ (-1)^{|τ_g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} $$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-4p})$.

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arXiv2018

Note on a sum involving the Euler function

We prove that $$ \sum_{n \leq x} φ([x/n])\leq\bigg(\frac{1380}{4009}+\frac{2629}{4009}\cdot\frac1{ζ(2)}+o(1)\bigg)x\log x $$ as $x\to\infty$, where $φ$ denotes the Euler totient function and $[x]$ denotes the integer part of $x$.

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arXiv2018

On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series

Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $α$ and $m\in\{1,2,3\}$, we have \begin{align*} \sum_{k=0}^{p^αn-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\equiv\bigg(\frac{m(m-4)}{p}\bigg)\sum_{k=0}^{p^{α-1}n-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\pmod{p^{2α}}, \end{align*} where $(x)_k=x(x+1)\cdots(x+k-1)$ and $\big(\frac{\cdot}{\cdot}\big)$ denotes the Legendre symbol. Also, when $m=4$, \begin{align*} \sum_{k=0}^{p^αn-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\equiv p\sum_{k=0}^{p^{α-1}n-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\pmod{p^{2α}}. \end{align*}

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arXiv2018

On a sum involving the Euler function

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.

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arXiv2018

On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$

In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$.

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arXiv2018

Divisibility of some binomial sums

With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $ρ,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^ρ\cdot (-4)^{ρ(n-1-k)} \equiv 0\pmod{2^{ρ-2}n\binom{2n}{n}}.$$

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arXiv2018

Supercongruences concerning truncated hypergeometric series

Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n}\\ &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv -Γ_p\bigg(\frac{1}{n}\bigg)^n\pmod{p^3}, $$ where the truncated hypergeometric series $$_nF_{n-1} \bigg[\begin{matrix} x_1&x_2&\ldots&x_n\\ &y_1&\cdots&y_{n-1}\end{matrix}\bigg | \, z\bigg]_m=\sum_{k=0}^{m}\frac{z^k}{k!}\prod_{j=0}^{k-1}\frac{(x_1+j)\cdots(x_{n}+j)}{(y_1+j)\cdots(y_{n-1}+j)} $$ and $Γ_p$ denotes the $p$-adic gamma function. This confirms a conjecture of Deines, Fuselier, Long, Swisher and Tu. Furthermore, under the same assumptions, we also prove that $$p^n\cdot {}_{n+1} F_n \bigg[ \begin{matrix} 1 &1 &\ldots &1\\ &\frac{n+1}{n} &\ldots &\frac{n+1}{n} \end{matrix}\bigg | \, 1\bigg]_{p-1} \equiv -Γ_p \Bigl(\frac{1}{n} \Bigr)^n \quad(\mathrm{mod}\ {p^3}),$$ which solves another conjecture of Deines, Fuselier, Long, Swisher and Tu.

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arXiv2018

On the sums of three generalized polygonal numbers

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$, $c$ for which the sum $P_a(x)+P_b(y)+P_c(z)$ represents all but finitely many positive integers.

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arXiv2018

On the divisibility of some truncated hypergeometric series

Let $p$ be an odd prime and $r\geq 1$. Suppose that $α$ is a $p$-adic integer with $α\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that $${}_{2r+1}F_{2r}\bigg[\begin{matrix}α&α&\ldots&α\\ &1&\ldots&1\end{matrix}\bigg|\,1\bigg]_{p-1}\equiv0\pmod{p^2},$$ where the truncated hypergeometric series $$ {}_{q+1}F_{q}\bigg[\begin{matrix}x_0&x_1&\ldots&x_{q}\\ &y_1&\ldots&y_q\end{matrix}\bigg|\,z\bigg]_{n}:=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_q)_k}{(y_1)_k\cdot (y_q)_k}\cdot\frac{z^k}{k!}. $$

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arXiv2018

On the gaps between consecutive primes

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.

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arXiv2016

On the number of three-term arithmetic progressions in a dense subset of $F_q^n$

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever $|A|\geq (c_q q)^n$ for some constant $c_q>0$. After the first version of our manuscript was uploaded in the arXiv, we learned from Professors Jacob Fox and Terence Tao that our result is a special case of a result of Fox and Lovasz [1, Theorem 3]. In fact, [1, Theorem 3] gives a much better bound than ours. For example, when $q=3$, the lower bound given by Fox and Lovasz is $|A|^{2}\cdot (|A|q^{-n})^{11.901}$, while our bound is $|A|^{2}\cdot (|A|q^{-n})^{25.803}$. We thank Professors Jacob Fox and Terence Tao for their helpful comments on our manuscript. [1] Jacob Fox, László Miklós Lovász, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230.

math.CO↗
arXiv2016

Supercongruences on some binomial sums involving Lucas sequences

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.

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arXiv2016

A generalization of Frieman's 3k-3 theorem

We prove a generalization of Frieman's $3k-3$ theorem for the sumset $$ Σ^{l}(A_1,\ldots,A_k)=\{a_{j_{1}}+\cdots+a_{j_{l}}:\,1\leq j_{1}<\cdots<j_{l}\leq k,\ a_{j_{s}}\in A_{j_{s}}\text{ for all }s\}. $$

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