It is well-known that the congruence $\sum_{i=1}^{ n} i^{ n} \equiv 1 \pmod{n}$ has exactly five solutions: $\{1,2,6,42,1806\}$. In this work, we characterize the solutions to the congruence $1^n + 2^n + \dotsb + n^n\equiv p \pmod{n}$ for every prime $p$. This characterization leads to an algorithm for computing all such solutions, when there is a finite number of them. More generally, our algorithm enables computing all the solutions below a much higher bound as compared to what can be achieved by a naive exhaustive search.
We show that if the congruence above holds and $n\mid m$, then the quotient $Q:=m/n$ satisfies $\sum_{p\mid Q} \frac{Q}{p}+1 \equiv 0\pmod{Q}$, where $p$ is prime. The only known solutions of the latter congruence are $Q=1$ and the eight known primary pseudoperfect numbers $2,6,42, 1806, 47058, 2214502422, 52495396602,$ and $8490421583559688410706771261086$. Fixing $Q$, we prove that the set of positive integers $n$ satisfying the congruence in the title, with $m=Q n$, is empty in case $Q=52495396602$, and in the other eight cases has an asymptotic density between bounds in $(0,1)$ that we pro…
In this paper we charatcterize primes of the form $x^2+dy^2$ with $x\equiv 0\pmod{N}$ or $y\equiv 0\pmod{N}$ for positive integer $N$ and $d$ with $d$ being square free.
In this paper we study the multiplicative function $ρ_{k,λ}(n)$ that counts the number of incongruent solutions of the equation $x_1^2+\cdots+x_k^2 \equiv λ\pmod{n}$. In particular we give closed explicit formulas for $ρ_{k,λ}(p^s)$ with a arithmetic complexity of constant order.
In the present paper we obtain several new results related to the problem of upper bound estimates for the number of solutions of the congruence $$ x^{x}\equiv λ\pmod p;\quad x\in \mathbb{N},\quad x\le p-1, $$ where $p$ is a large prime number, $λ$ is an integer corpime to $p$. Our arguments are based on recent estimates of trigonometric sums over subgroups due to Shkredov and Shteinikov.
Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$, then the Hilbert class field of $\K$ is $\H_\K=\K(\sqrt{2})$. Finally, we find a criterion to decide if an ideal of the ring of integers or $\K$ is principal or non-principal.
In this paper we consider the congruence equation $q_1 q_2 \equiv c \pmod q$ with $a < q_1 \leq a + q^{1/2+ε}$ and $b < q_2 \leq b + q^{1/2+ε}$ and show that it has solution for almost all $a$ and $b$. Then we apply it to a question of Fujii and Kitaoka as well as generalize it to more variables. At the end, we will present a new way to attack the above congruence equation question through higher moments.
The multiplicative structure of an odd perfect number $n$, if any, is $n=π^αM^2$, where $π$ is prime, $\gcd(π,M)=1$ and $π\equiv α\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv π\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $π$). We further extend the proof to $x=α+2$, $α+2$ prime, with the restri…
Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such congruences, those of the form $p_{-t}(\ell n + a) \equiv 0 \pmod {\ell}$, were previously studied by Kiming and Olsson. If $\ell \geq 5$ is prime and $-t \not \in \{\ell - 1, \ell -3\}$, then such congruences satisfy $24a \equiv -t \pmod {\ell}$. Inspired by Lin's example, we obtain natural infinite families of such congruences. If $\ell\equiv2\pmod{3}$ (resp. $\ell…
We build a new theory for analyzing the coefficients of any cyclotomic polynomial by considering it as a gcd of simpler polynomials. Using this theory, we generalize a result known as periodicity and provide two new families of flat cyclotomic polynomials. One, of order 3, was conjectured by Broadhurst: $Φ_{pqr}(x)$ is flat where $p<q<r$ are primes and there is a positive integer $w$ such that $r\equiv\pm w\pmod{pq}$, $p\equiv1\pmod w$ and $q\equiv1\pmod{wp}$. The other is the first general family of order 4: $Φ_{pqrs}(x)$ is flat for primes $p,q,r,s$ where $q\equiv-1\pmod p$, $r\equiv\pm1\pmo…
Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle binomial coefficient to a certain power of two. The striking thing about Morley's congruence is that it is valid modulo the third power of the prime being considered.
In this work we consider the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1 \pmod n$ for each $k \in \mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ satisfies this congruence if and only if $n$ is prime or a Giuga Number and $λ(n) \mid k(n-1)$. In passing, we establish new characterizations of Giuga numbers and we study some properties of the numbers $n$ satisfying $λ(n) \mid k(n-1)$.
In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or} \quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positive rational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently, N. X. Tho proved the following statement: let $a\in\mathbb N$ be odd and let either $n\equiv 0\pmod 4$ or $n\equiv 7\pmod 8$. Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers $x,y,z.$ A representative example of our result is the following statement: assume that $a,n\in\mathbb N$ are such that at least on…
For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.
Let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of weight $n$. Recently, Hirschhorn and Sellers, Yao, and Xia established a number of congruences modulo 2 and 5, 4 and 8, and 25 for $f(n)$, respectively. In this note, we shall prove several new congruences modulo 125 and 11 by using some results of modular forms. For example, for all $n\ge 0$, we have \begin{align*} f(1250n+125)&\equiv 0 \pmod{125},\\ f(1250n+1125)&\equiv 0 \pmod{125},\\ f(2750n+825)&\equiv 0 \pmod{11},\\ f(2750n+1925)&\equiv 0 \pmod{11}. \end{align*}
Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively prime with $4q$ (modulo $4q$) such that $x^2\equiv q\pmod{p}$ has a solution iff $p\equiv c\pmod{4q}$ for some $c\in\mathbb{L}_{4q}$. Moreover, $\mathbb{L}_{4q}$ is the only subgroup of $\mathbb{U}_{4q}$ of half order containing $-1$. Considering the ring $\mathbb{Z}[\sqrt{2}]$, for any odd prime $p$ it is known that the equation $x^2\equiv 2\pmod{p}$ has a solu…
Let $d$ be a square-free integer, $\mathbf{k}=\mathbb{Q}(\sqrt d,\,i)$ and $i=\sqrt{-1}$. Let $\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}$, $\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}_1^{(2)}$ and $G=\mathrm{Gal}(\mathbf{k}_2^{(2)}/\mathbf{k})$ be the Galois group of $\mathbf{k}_2^{(2)}/\mathbf{k}$. Our goal is to give necessary and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with $p$ and $q$ are primes such that $p\equiv 1\pmod 8$ and $q\equiv 5\pmod 8$ or $p\equiv 1\pmod 8$ and $q\equiv 3\pmod 4$.
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0 \pmod{4}$, $\overline{p}(4n+3)\equiv 0 \pmod{8}$ and $\overline{p}(8n+7)\equiv 0 \pmod{64}$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8, 16$ and $32$, such as $\overline{p}(48n+26) \equiv 0 \pmod{8}$, $\o…
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-\f…
Suppose that $d \in \{ 2, 3, 4, 6 \}$ and $a \in \mathbb{Z}$ with $a\neq -1$ and $a$ is not square. Let $P_{(a,d)}$ be the number of primes $p$ not exceeding $x$ such that $p \equiv 1 \pmod{d}$ and $a^{(p-1)/d} \equiv 1 \pmod{p}$. In this paper, we study the mean value of $P_{(a,d)}$.
Let $p\equiv3\pmod{4}$ be a prime and $r$ a positive integer. We show that $$ \prod_{k=1}^{(p^{2r}-1)/2}\frac{4k-1}{4k+1}\equiv1\pmod{p^2}. $$ This confirms a recent conjecture of Guo.
Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\equiv (-1)^{\langle a\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\pmod{p^3}$$ and for $a\not\equiv -\frac 12\pmod p$, $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k+1}\equiv \frac{1+2t}{1+2a} +p^2\frac{t(t+1)}{1+2a}B_{p-2}(-a)\pmod{p^3},$$ where $\langle a\rangle_p\in\{0,1,\ldots,p-1\}$ satisfying $a\equiv \langle a\rangle_p\pmod p$ and $t=(a-\langle a\rangle_p)/p$. Tak…
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 $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \[\begin{cases} \begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\equiv0\pmod{(m^2+n^2)^2},\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\equiv0 \pmod{(m^2+n^2)^2}, \end{split} \end{cases}\] where $k\equiv0\pmod{4}$ when…
Euler showed that if an odd perfect number exists, it must be of the form $N = p^αq_{1}^{2β_{1}}$ $\ldots$ $q_{k}^{2β_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $α$, $β_{i} \geq 1$, for $1 \leq i \leq k$, with $p \equiv α\equiv 1 \pmod{4}$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $β_{i} \equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each $β_{i} \equiv 2 \pmod{5}$.
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that $\sum_{k=0}^{\frac{p-1}{2}}\binom{2k}k^3\equiv 0\pmod {p^2}$ for $p\equiv 3,5,6\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\frac {p}{4}]}(t)\equiv -\big(\frac{-6}{p}\big)\sum_{x=0}^{p-1} \big(\frac{x^3-3/2(3t+5)x-9t-7}{p}\big)\pmod p$ and d…
Let $Φ_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]Φ_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the…
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 $x,y\in\Bbb Z$.
We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv 0\pmod p$, then the congruence $$ ax_1\cdots x_6+bx_7\cdots x_{13}\equiv c\pmod p $$ has a solution with $x_j\in\cI_j$. There exists an absolute constant $n_0\in\N$ such that for any $0<\varepsilon<1$ and any sufficiently large prime $p$, any quadratic residue $λ$ modulo $p$ can be represented in the form $$ x_1\cdots x_{n_0}\equiv λ\pmod p,\quad x_i\in\N,\quad x…
Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$. Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $t a \equiv \pm \tilde{a} \pmod{p}.$ For any two such pairs, assume $a<\tilde{a}, b<\tilde{b}, a<b $, then there are three possibilities for their relative order : $a<\tilde{a} < b< \tilde{b}$ , $a< b < \tilde{a} < \tilde{b}$ , $a< b < \tilde{b}< \tilde{a}$. We show this paring is balanced in the sense that the three cases occur with equal frequencies. Utilizing prope…
Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+β) \equiv 0 \pmod{\ell}.$ Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$. In this paper, for a wide range of $c \in \mathbb{F}_{\ell}$, we prove congruences of the form $p(\ell Q^{3} n+β_{0}) \equiv c \cdot p(\ell Q n+β_{1}) \pmod{\ell}$ for infinitely many primes $Q$. For a positive integer $r$, let $p_{r}(n)$ be the $r$-colored par…
Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to show that any $λ\not\equiv 0\pmod p$ can be represented in the form $λ\equiv n_1!...n_7!\pmod p$, where $n_i=o(p^{11/12})$. This slightly refines the previously known range for $n_i$.
In this paper, we show that if $p\equiv 1\pmod 4$ is prime, then $4F_p$ admits a representation of the form $u^2-pv^2$ for some integers $u$ and $v$, where $F_n$ is the $n$th Fibonacci number. We prove a similar result when $p\equiv -1\pmod 4$.
Let $\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge 0$, \[\mathrm{pod}(3n+2)\equiv 2{{(-1)}^{n+1}}{{r}_{5}}(8n+5) \pmod{9}, \] and \[\mathrm{pod}(5n+2)\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3) \pmod{5}.\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \in \{11, 19\}$ and any integers $α\ge 1$ and $n \ge 0$, we have \[\mathrm{pod}\B…
We give a new proof of a theorem by P. Mihailescu which states that the equation $x^p-y^q=1$ is unsolvable with $x, y$ integral and $p, q$ odd primes, unless the congruences $p^q \equiv p\pmod{q^2}$ and $q^p\equiv q \pmod{p^2}$ hold.
In this paper we study Gaussian ring $\Z[i]$ with a focus on representing Gaussian Mersenne primes $G_p$ in the form $x^2+7y^2$. Interestingly when such a form exists, one can observe that, $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$. To prove this property of Gaussian Mersenne primes, we show that Gaussian Mersenne primes splits completely in the cyclic quartic unramified extension of $\Q(\sqrt{-14})$ and have a trivial Artin symbol in this extension. We generalize this result for $d\equiv 7\pmod{24}$. We also attempt to give an alternate proof using Artin's reciprocity law, which was ear…
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0 \pmod{8}$ for $n\geq 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0 \pmod{40}$ for $n\geq 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\overline{p}(5n)\equiv (-1)^{n}\overline{p}(4\cdot 5n) \pmod{5}$ for $n \geq 0$ and $\overline{p}(n)\equiv (-1)^{n}\overline{p}(4n)\pmod{8}$ for $n \geq 0$ by using the relation of the generating function o…
We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.
In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.
Let $p$ be a prime number with $p>3$, $p\equiv 3\pmod{4}$ and let $n$ be a positive integer. In this paper, we prove that the Diophantine equation $(5pn^{2}-1)^{x}+(p(p-5)n^{2}+1)^{y}=(pn)^{z}$ has only the positive integer solution $(x,y,z)=(1,1,2)$ where $pn \equiv \pm1 \pmod 5$. As an another result, we show that the Diophantine equation $(35n^{2}-1)^{x}+(14n^{2}+1)^{y}=(7n)^{z}$ has only the positive integer solution $(x,y,z)=(1,1,2)$ where $n\equiv \pm 3% \pmod{5}$ or $5\mid n$. On the proofs, we use the properties of Jacobi symbol and Baker's method.
Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big…
Let $q$ be an odd prime and $f(x)$, $g(x)$ be polynomials with integer coefficients. If the system of congruences $f(x) \equiv g(x) \equiv 0 \pmod{q}$ has $\ell$ solutions, then $R\left(f(x),g(x)\right)\equiv 0 \pmod{q^\ell}$, where $R\left(f(x),g(x)\right)$ is the resultant of the polynomials. Using this result we give new proofs of some known congruences involving the Lucas sequences.
In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+d}}{64^k}\equiv 0\pmod{p^2}$$ for all $d\in\{0,\ldots,p-1\}$ with $d\equiv (p+1)/2\pmod2$. If $p\equiv 1\pmod4$ and $p=x^2+y^2$ with $x\equiv 1\pmod4$ and $y\equiv 0\pmod2$, then we show $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+1}}{(-8)^k}\equiv 2p-2x^2\pmod{p^2}\ \ \mbox{and}\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{2k}{k+1}^2}{(-8)^k}\equiv-2p\pmod{p^2}…
The Apéry polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Apéry numbers.) Let $p$ be an odd prime. We show that $$\sum_{k=0}^{p-1}(-1)^kA_k(x)\equiv\sum_{k=0}^{p-1}\frac{\binom{2k}k^3}{16^k}x^k\pmod{p^2},$$ and that $$\sum_{k=0}^{p-1}A_k(x)\equiv\left(\frac xp\right)\sum_{k=0}^{p-1}\frac{\binom{4k}{k,k,k,k}}{(256x)^k}\pmod{p}$$ for any $p$-adic integer $x\not\equiv 0\pmod p$. This enables us to determine explicitly $\sum_{k=0}^{p-1}(\pm1)^kA_k$ mod $p$, and $\sum_{k=0}^{p-1}(-1)^kA_k$ mod $p^2$ in the case $p\equiv 2\pmo…
In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{4k}{2k+1}\binom{2k}k}{48^k}\equiv0\pmod{p^2},$$ and $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv1\pmod{3},\\ p/\binom{(2p+2)/3}{(p+1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv2\pmod{3}.\end{cases} $$ We also obtain some other results of such types.
Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2α}n+\frac{3^{2α}-1}{4}\right )\equiv 0 \pmod{3^α}\] and \[a_9\left (3^{α+1}n+3^{α+1}-1\right )\equiv 0 \pmod{3^{α+1}}.\]
We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmodα$ has the solution $β$ with $0 < β< α$ if and only if $α/β$ has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number $α/β> 1$ in lowest terms has a symmetric continued fraction precisely when $β^2 \equiv \pm 1 \pmodα$.
Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide unless $p\equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$.
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 co…
Let $ε$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+ε},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j] \} $$ contains almost all the residue classes modulo $m$ (i.e., its cardinality is equal to $m+o(m)$). We further show that if $m$ is cubefree, then for any positive integers $N_1,N_2,N_3,N_4$ with $N_1N_2N_3N_4>m^{1+ε},$ the set $$ \{x_1x_2x_3x_4 \pmod m: \quad x_j\in [1,N_j] \} $$ also contains almost all the residue classes modulo $m.$ Let $p$ be a large prime paramet…
We use the arithmetic of ideals in orders to parameterize the roots $μ\pmod m$ of the polynomial congruence $F(μ) \equiv 0 \pmod m$, $F(X) \in \mathbb{Z}[X]$ monic, irreducible and degree $d$. Our parameterization generalizes Gauss's classic parameterization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial $F(X) = X^3 - 2$. We show that only a special class of ideals are needed to parameterize the roots $μ\pmod m$, and that in the cubic setting, $d = 3$, general ideals correspond to pairs of roots $μ_1 \pmod{m_1…
The cubic partitions of a natural number $n$, introduced by Chan and Kim, have generating function $\sum_{n=0}^{\infty}a(n)q^n= \frac{1}{(q; q)_{\infty}(q^2; q^2)_{\infty}}.$ In this paper, we generalize some results of Chen-Lin, which suggest that $a(n)$ should have analogous properties of the ordinary partition function. Specifically, we show that for every non-negative integer $n$, $a(5^4n+547)\equiv 0\pmod{5^2}, a(7^3n+190)\equiv 0\pmod{7^2}, a(7^3n+288 \equiv 0\pmod{7^2} and a(7^3n+337)\equiv 0\pmod{7^2}.$
