若$a>b$、$n>1$为正整数且$a$与$b$互素，则$(a,b,n)$的大Zsigmondy素数是满足$p \mid a^n-b^n$、但对$1 \leq m < n$有$p \nmid a^m-b^m$，且要么$p^2 \mid a^n - b^n$要么$p > n + 1$的素数$p$。我们对所有不存在大Zsigmondy素数的整数三元组$(a, b, n)$进行了分类。
英文摘要：
If $a>b$ and $n>1$ are positive integers and $a$ and $b$ are relatively prime integers, then a large Zsigmondy prime for $(a,b,n)$ is a prime $p$ such that $p \,|\, a^n-b^n$ but $p \,\nmid \, a^m-b^m$ for $1 \leq m < n$ and either $p^2 \, | \, a^n - b^n$ or $ p > n + 1$. We classify all the triples of integers $(a, b, n)$ for which no large Zsigmondy prime exists.
In the number field context, Bang-Zsigmondy's theorem states that for any integers $u, m > 1$, there exists a prime divisor $p$ of $u^m - 1$ such that $p$ does not divide $u^n - 1$ for every integer $0 < n < m$ except in some exceptional cases that can be explicitly determined. A prime $p$ satisfying the conditions in Bang-Zsigmondy's theorem is called a Zsigmondy prime for $(u, m)$. In 1988, Feit introduced the notion of large Zsigmondy primes as follows: A Zsigmondy prime $p$ for $(u, m)$ is called a large Zsigmondy prime if either $p > m + 1$ or $p^2$ divides $u^m - 1$. In the same year, Fe…
For an integer $a$ consider the sequence $(T_{n}(a)-1)_{n=1}^{\infty}$ defined by the Chebyshev polynomials $T_{n}$. We list all pairs $(n,a)$ for which the term $T_{n}(a)-1$ has no primitive prime divisor.
We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor. For the sequences $k^u+h^u$ and $k^u-h^u$, we discern a formula to find the primitive indexes of any composite number given the primitive indexes of its prime factors. We show how this formula reduces to a formula relating the multiplicative order of $k$ modulo $N$ to that of its prime factors. We then introduce immediate consequences of the formula: certain sequences which yield the same primitive indexes for numbers with the same unique prime factors, an expansion of t…
Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $β\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.
A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime divisors. Let f : X --> X be a self-morphism of a variety, let D be an effective divisor on X, and let P be a point of X, all defined over the algebraic closure of Q. We consider the Zsigmondy set Z(X,f,P,D) of the sequence defined by the arithmetic intersection of the f-orbit of P with D. Under various assumptions on X, f, D, and P, we use Vojta's conjecture wit…
We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical orbits, the Zsigmondy set is explicitly computed by utilizing effective dynamical height bounds. In the general case, we use Thue-style Diophantine approximation methods to bound the size of the Zsigmondy set when d >2, and complex-analytic methods when d=2.
In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{cases}-(A^2+4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=1, \\(-A^2-4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2+5ij+2j^2…
Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ with $gcd(a,b)=1$, $p \nmid ab$, and $p|(a^n \pm b^n)$, we prove the identity $$ν_p(a^n \pm b^n)-ν_p(n)=ν_p(a^{p-1}-b^{p-1}).$$ An unintended interesting immediate consequence is the following variant of Wieferich's criterion for FLT : Let $x^n+y^n=z^n$ with $n$ prime and $x,y,z$ pairwise relatively prime. Then every odd prime $p|y$ satisfies $ν_p(z^{p-1}-x^{p-1}) \ge n-1$ and every odd prime $p|x$ satisfies $ν_p(z^{p-1}-y^{p-1}) \ge n-1$, and every odd prime $p|z$ satisfies $ν_p(x^{p-1}-y^{p-1}) \ge n-1$, ie. every …
Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $λ(a,b;n)$ be given by $q\prod_{k=1}^\infty (1-q^{ak})^3(1-q^{bk})^3 = \sum_{n=1}^\infty λ(a,b;n)q^n$. In the paper, using Jacobi's identity $\prod_{n=1}^\infty (1-q^n)^3 = \sum_{k=0}^\infty (-1)^k(2k+1)q^{\frac{k(k+1)}2}$ we construct $x^2$ in terms of $λ(a,b;n)$. For example, if $2\nmid ab$ and $p\nmid ab(ab+1)$, then $(-1)^{\frac{a+b}2x+\frac{b+1}2}(4ax^2-2p) = λ(a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $λ(1,3;n+1),λ(1,7;2n+1)$, $λ(3,5;2n+1)$ and $λ(1,15;4n+1…
Let $D$ be a negative integer congruent to $0$ or $1\bmod{4}$ and $\mathcal{O}=\mathcal{O}_D$ be the corresponding order of $ K=\mathbb{Q}(\sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $ j_D=j(\mathbb{C}/\mathcal{O})$ of $\mathcal{O}$ over $K$. Let $n_D=(\mathcal{O}_{\mathbb{Q}( j_D)}:\mathbb{Z}[ j_D])$ denote the index of $\mathbb{Z}[ j_D]$ in the ring of integers of $\mathbb{Q}(j_D)$. Suppose $p$ is any prime. We completely determine the factorization of $H_D(x)$ in $\mathbb{F}_p[x]$ if either $p\nmid n_D$ or $p\nmid D$ is inert in $K$ and t…
Let $N$ be an odd perfect number. Let $ω(N)$ be the number of distinct prime factors of $N$ and let $Ω(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}ω(N) - \frac{187}{37} \leq Ω(N)$ and that if $3\nmid N$, then $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.
Let $ϕ$ denote a primitive Hecke-Maass cusp form for $Γ_o(N)$ with the Laplacian eigenvalue $λ_ϕ=1/4+t_ϕ^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|α_{p}|=|β_{p}| = 1$, and $p\ll(N(1+|t_ϕ|))^c$, where $α_{p},\;β_{p}$ are the Satake parameters of $ϕ$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|α_{p}|=|β_{p}| = 1$) is at least $34/35$.
A prime number $p$ is called a Schenker prime if there exists such $n\in\mathbb{N}_+$ that $p\nmid n$ and $p\mid a_n$, where $a_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\in\mathbb{N}_+$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ h…
Let $\Bbb Z$ be the set of integers, and let $p$ be a prime of the form $4k+1$. Suppose $q\in\Bbb Z$, $2\nmid q$, $p\nmid q$, $p=c^2+d^2$, $c,d\in\Bbb Z$ and $c\equiv 1\pmod 4$. In this paper we continue to discuss congruences for $q^{[p/8]}\pmod p$ and present new reciprocity laws, but we assume $4p=x^2+qy^2$ or $p=x^2+2qy^2$, where $[\cdot]$ is the greatest integer function and $x,y\in\Bbb Z$.
Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.
For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient num…
For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.
Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \nmid D$, and $0< σ< 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,σ)}$, then there exists an effectively computable constant $C_p$ such that for every $x> C_P$ with $x^2+D=p^n.m $, we have $ m > x^σ$. As an application, we show that for $x \neq \{1015,5 \}$, if the equation $x^2+76=101^n.m $ holds, we have $ m > x^{0.14}$. .
Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation $$cx^2+p^{2m}=4y^n,~~x,y\geq 1, m\geq 0, n\geq 3, \gcd(x,y)=1, \gcd(n,2h(-c))=1,$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.
Let $\mathbf{u} = (u_n)_{n \geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \geq 2$, where $a_1$ and $a_2$ are fixed nonzero integers. For each prime number $p$ with $p \nmid 2a_2D_{\mathbf{u}}$, where $D_{\mathbf{u}} := a_1^2 + 4a_2$, let $ρ_{\mathbf{u}}(p)$ be the rank of appearance of $p$ in $\mathbf{u}$, that is, the smallest positive integer $k$ such that $p \mid u_k$. It is well known that $ρ_{\mathbf{u}}(p)$ exists and that $p \equiv \big(D_{\mathbf{u}} \mid p \big) \pmod {ρ_{\ma…
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{3k}km^{-k}\mod {p^2}$.
The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is also an integer; we call it the Lerch quotient $\ell_p.$ If $p\mid\ell_p$ we say $p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to $3\times10^6$; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if $p$ is a non-Wilson prime, then $q_p(w…
Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\ldots$ such that for sufficiently large primes $p$ we have $\sum_{k=0}^{p-1}a_k\equiv x^2-2p$ (mod $p^2$) if $4p=x^2+dy^2$ (and $4\nmid x$ if $d=1$), and $\sum_{k=0}^{p-1}a_k\equiv 0$ (mod $p^2$) if $(\frac{-d}p)=-1$. In this paper we give a survey of conjectures and results on this topic and point out the connection between this problem and series for $1/π$.
Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level $N$, trivial nebentypus, and no complex multiplication (CM). For all primes $p$, we may define $θ_p\in [0,π]$ such that $a_f(p) = 2p^{(k-1)/2}\cos θ_p$. The Sato-Tate conjecture states that the angles $θ_p$ are equidistributed with respect to the probability measure $μ_{\textrm{ST}}(I) = \frac{2}π\int_I \sin^2 θ\; dθ$, where $I\subseteq [0,π]$. Using recent results on the automorphy of symmetric-power $L$-functions due to Newton and Thorne, we explicitly bound the error term in…
Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every sufficiently large prime $p$ $$ \sum_{\substack{l_1+l_2+\cdots+l_n=mp^r p\nmid l_1 l_2 \cdots l_n }} \frac1{l_1l_2\cdots l_n} \equiv p^{r-1} \sum_{{\bf k}\vdash n} C_{m,{\bf k}} B_{p-{\bf k}} \pmod{p^r} $$ where $B_{p-{\bf k}}=B_{p-k_1}B_{p-k_2}\cdots B_{p-k_t}$ are products of Bernoulli numbers and the coefficients $C_{m,{\bf k}}$ are polynomials of $m$ independent of…
In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $$\sum_{k=0}^{p-1}\frac{(q-\frac{p}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3} $$ and $$p^n\sum_{k=0}^{p-1}\frac{(1)_k^n}{(\frac{p}{n}-q+2)_k^n}\equiv0\pmod{p^3}$$ for any prime $p>\max\{n,(q-1)n+1\}$, where $(x)_k$ denotes the Pochhammer symbol defined by $$ (x)_k=\begin{cases}1,\quad &k=0,\\ x(x+1)\cdots(x+k-1),\quad &k>0.\end{cases}$$ Let $n\geq4$ be an even integer. Then for a…
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we prove some supercongruences concerning $$\align &\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k}{54^k},\ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{4k}{2k}}{128^k},\ \sum_{k=0}^{p-1}\frac{\binom{3k}k\binom{6k}{3k}}{432^k}, &\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{3k}{k}}{m^k}, \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k},\ \sum_{k=0}^{p-1}\f{\binom{2k}k\binom{3k}{k}\binom{6k}{3k}}{m^k}\mod {p^2}.\endalign$$ Thus we solve some conjectures of Zhi-Wei Sun and the author.
For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that there is no infinite subset $\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+\cdot +a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\ a_{i,j}\in \mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Rué for the bilinear case, and answers a question pos…
We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ν+1$, and $K_d := \mathbb{F}_p(μ_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they g…
For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that there is no infinite subset $\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+...+a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\ a_{i,j}\in \mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Rué for bilinear case, and answers a question posed by S…
