Recently, Z.-W. Sun made the following conjecture: for any odd prime $p$ and odd integer $m$, $$ \frac{1}{m^2{m-1\choose (m-1)/2}}\Bigg(\sum_{k=0}^{(pm-1)/2}\frac{2k\choose k}{8^k} -\left(\frac{2}{p}\right)\sum_{k=0}^{(m-1)/2}\frac{2k\choose k}{8^k}\Bigg) \equiv 0\pmod{p^2}. $$ In this note, applying the "creative microscoping" method, introduced by the author and Zudilin, we confirm the above conjecture of Sun.
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \end{align*} In fact, this supercongruence is a generalization of a supercongruence of van Hamme.
In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences. All of the 13 supercongruences have been confirmed by using a wide range of methods. In 2015, Swisher conjectured Dwork-type supercongruences related to the first 12 supercongruences of Van Hamme. Here we prove that the (C.3) and (J.3) supercongruences of Swisher are true modulo $p^{3r}$ (the original modulus is $p^{4r}$) by establishing $q$-analogues of them. Our proof will use the creative microscoping method, recently introduced by the author in collaboration with Zudilin. We also raise conjectures on $q$-analogues of an equ…
In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to supercongruences modulo $p^4$ for primes $p\equiv 3\pmod{4}$, which appear to be new.
The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv 0\pmod{n{2n\choose n}}, \\ \sum_{k=0}^{p-1}\frac{3k+1}{(-16)^k} {2k\choose k} f_k &\equiv p (-1)^{\frac{p-1}{2}} \pmod{p^3}. \end{align*} where $n>1$ and $p$ is an odd prime. The second congruence modulo $p^2$ confirms a recent conjecture of Z.-H. Sun. We also show that, if $p$ is a prime of the form $4k+3$, then $$ \sum_{k=0}^{p-1}\frac{{2k\choose k} f_k}{(-16)^k} \equiv 0 …
By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\…
We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.
We develop an analytical method to prove congruences of the type $$ \sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv ω(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]} \quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers $m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic" hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in 1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of several Dwork-type congruences corresponding to $m\ge2$ (in other words, supercongruences) are based on constructing and proving t…
In this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in 2013. Let $p$ be an odd prime and let $a\in\mathbb{Z}^{+}$. Then if $p\equiv1\pmod3$, we have \begin{align*} \sum_{k=0}^{\lfloor\frac{5}6p^a\rfloor}\frac{\binom{2k}k}{16^k}\equiv\left(\frac{3}{p^a}\right)\pmod{p^2}, \end{align*} where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol.
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv (2q+2q^{-1}-1)[n]_{q^2}^4\pmod{[n]_{q^2}^4Φ_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ for $k\geq 1$ and $Φ_n(q)$ denotes the $n$-th cyclotomic polynomial.
Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross-Koblitz formula. In this paper, we aim to prove four supercongruences between two truncated hypergeometric series ${}_2F_1$. This generalizes these four Rodriguez-Villegas supercongruences.
In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version…
We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $α$ is $p$-integral, then we can completely determine $$ \sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k}\quad\text{and}\quad \sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k} $$ modulo $p^3$. In particular, by setting $α=-1/2$, we confirm two conjectured congruences of Z.-W. Sun.
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.
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials.
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.
In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.
In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.
The Delannoy numbers and Schröder numbers are given by \begin{align*} D_n=\sum_{k=0}^n{n\choose k}{n+k\choose k}\quad \text{and}\quad S_n=\sum_{k=0}^n{n\choose k}{n+k\choose k}\frac{1}{k+1}, \end{align*} respectively. Let $p>3$ be a prime. We mainly prove that \begin{align*} \sum_{k=1}^{p-1}D_k S_k\equiv 2p^3B_{p-3}-2pH^{*}_{p-1} \pmod{p^4}, \end{align*} where $B_n$ is the $n$-th Bernoulli number and those $H^{*}_n$ are the alternating harmonic numbers given by $H^{*}_n=\sum_{k=1}^{n}\frac{(-1)^k}{k}$. This supercongruence was originally conjectured by Z.-W. Sun in 2011.
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ \sum_{k=0}^{p-1}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supe…
The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences: $$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\ \mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\ \mbox{modulo}\ p$$ for any prime $p>3$, the second one was conjectured by Z.-W. Sun in 2012. These two congruences are very important to prove the following conjectures of Z.W.Sun: For any old prime $p$, we have $$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\bi…
In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.
In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $p-$adic gamma function, and are called Ramanujan type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the prove…
The Apéry numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=\sum_{k=0}^{n}{\binom{n+k}{2k}}^2{\binom{2k}{k}}^2\ \ \ \ \ {\rm and }\ \ \ \ \ \ f_n=\sum_{k=0}^{n}{\binom{n}{k}}^3(n=0, 1, \cdots,).$$ In this paper, we prove three supercongruences for Apéry numbers or Franel numbers conjectured by Z.-W. Sun. Let $p\geq 5$ be a prime and let $n\in \mathbb{Z}^{+}$. We show that \begin{align} \notag \frac{1}{n}\bigg(\sum_{k=0}^{pn-1}(2k+1)A_k-p\sum_{k=0}^{n-1}(2k+1)A_k\bigg)\equiv0\pmod{p^{4+3ν_p(n)}} \end{align} and \begin{align}\notag \frac{1}{n^3}\bigg(\sum_{k=0}^{pn-1}(2k+1)^3A_k-p^…
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…
