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Register a new userProof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial
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By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Phi_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
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We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.
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.
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}equiv (2q+2q^{-1}-1)[n]_{q^2}^4pmod{[n]_{q^2}^4Phi_n(q^2)}, $$ 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 $kgeq 1$ and $Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.
Let $f(x)inmathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $nge 1$ and $kge 2$. An integer $a$ is called an $f$-exunit in the ring $mathbb{Z}_n$ of residue classes modulo $n$ if $gcd(f(a),n)=1$. In this paper, we use the principle of cross-classification to derive an explicit formula for the number ${mathcal N}_{k,f,c}(n)$ of solutions $(x_1,...,x_k)$ of the congruence $x_1+...+x_kequiv cpmod n$ with all $x_i$ being $f$-exunits in the ring $mathbb{Z}_n$. This extends a recent result of Anand {it et al.} [On a question of $f$-exunits in $mathbb{Z}/{nmathbb{Z}}$, {it Arch. Math. (Basel)} {bf 116} (2021), 403-409]. We derive a more explicit formula for ${mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.
The scaled inverse of a nonzero element $a(x)in mathbb{Z}[x]/f(x)$, where $f(x)$ is an irreducible polynomial over $mathbb{Z}$, is the element $b(x)in mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c pmod{f(x)}$ for the smallest possible positive integer scale $c$. In this paper, we investigate the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $Phi_{p^s}(x)$ or $Phi_{p^s q^t}(x)$, where $p, q$ are primes with $p<q$ and $s, t$ are positive integers. Our main results are that the coefficient size of the scaled inverse of $(x^i-x^j)$ is bounded by $p-1$ with the scale $p$ modulo $Phi_{p^s}(x)$, and is bounded by $q-1$ with the scale not greater than $q$ modulo $Phi_{p^s q^t}(x)$. Previously, the analogous result on cyclotomic polynomials of the form $Phi_{2^n}(x)$ gave rise to many lattice-based cryptosystems, especially, zero-knowledge proofs. Our result provides more flexible choice of cyclotomic polynomials in such cryptosystems. Along the way of proving the theorems, we also prove several properties of ${x^k}_{kinmathbb{Z}}$ in $mathbb{Z}[x]/Phi_{pq}(x)$ which might be of independent interest.
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