27 problems

Let ${\mathbb Z}^{+}$ denote the positive integers, and for $n\in{\mathbb Z}^{+}$ define … where … Write $w_n(x)=w_n^{(1)}(x)$, let $(a,b)$ denote the greatest common divisor of…

Let $m$ and $n$ be positive integers with $n$ odd, and let $(a;q)_k$ denote the q-shifted factorial. The cubic q-supercongruence conjecture. … The source records this as another si…

Let $m$ and $n$ be positive integers with $n$ odd, and let $(a;q)_k$ denote the q-shifted factorial. The cubic q-supercongruence conjecture. … The $m=1$ case was proved by the auth…

Let $m$ and $n$ be positive integers with $n\equiv1\;\allowbreak(\operatorname{mod}4)$, and let $(a;q)_k$ denote the q-shifted factorial. The fifth-order q-supercongruence conjectu…

Let $n\equiv1\;\allowbreak(\operatorname{mod}4)$ be an integer greater than $1$, and let $r,s$ be positive integers with $r>s$. For $(a;q)_k$ the q-shifted factorial, the factorial…

Let $n$ and $s$ be the parameters occurring in Lemma 3, and let $\Phi_{n^s}(q)$ be the corresponding cyclotomic polynomial. The strengthening conjecture. The q-congruence in Lemma…

Let $m$ and $n$ be positive integers with $n\equiv1\;\allowbreak(\operatorname{mod}4)$ and $n>1$, and let $(a;q)_k$ denote the $q$-shifted factorial. The q-analogue conjecture. … T…

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 8 by … Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polyn…

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 7. Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomia…

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 4. Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomia…

Let $n$ be a positive odd integer, and let $a$ be an indeterminate. Write $[n]$ for the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. Let the congruences…

Let $n$ be a positive integer and let $r$ be an arbitrary integer. Write $[n]=(1-q^n)/(1-q)$ for the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. For int…

Define the $q$-shifted factorial by $(a;q)_0=1$ and $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ for $k\geq 1$, let $\Phi_n(q)$ be the $n$-th cyclotomic polynomial, and write…

Define the $q$-shifted factorial by $(a;q)_0=1$ and $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ for $k\geq 1$, and let $\Phi_n(q)$ denote the $n$-th cyclotomic polynomial. Let $m$ and…

Let $(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)$ be the $q$-shifted factorial, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and let $n>1$ be odd and $r\geqslant 1$. Partial q-analogu…

Let $(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)$ be the $q$-shifted factorial, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and let $n>1$ be odd and $r\geqslant 1$. Guo–Zudilin's hal…

Let $(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)$ be the $q$-shifted factorial, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and let $n>1$ be odd and $r\geqslant 1$. Guo–Zudilin's ful…

Let $d\geqslant 3$ and let $n>1$ be an integer with $n\equiv 1\pmod{d}$. With $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$, $[m]=1+q+\cdots+q^{m-1}$, and $\Phi_n(q)$ the $n$-th cyclotom…

Let $d\geqslant 3$ and let $n$ be a positive integer with $n\equiv -1\pmod{d}$. Let $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ denote the $q$-shifted factorial, let…

The conjecture. Then

The conjecture. The truncated series should satisfy

Let $n$ be a positive integer with $n\equiv 3\pmod{4}$, let $r$ be any positive integer, and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. Then the conjectured q-congruen…

Tauraso's conjecture.

Let the two preceding results, Theorems $\mathrm{thm:fatorodd\text{-}first}$ and $\mathrm{thm:fatorodd\text{-}second}$, be the stated congruence theorems involving the parameter…

General $q$-congruence conjecture. One has