Quantum mechanics

Pure mathematics

Discrete mathematics

Prime (order theory)

Power (physics)

Polynomial

Physics

Mathematics

Chinese remainder theorem

Mathematical analysis

Combinatorics

Conjecture

Modulo

Generalization

Cyclotomic polynomial

Primitive root modulo n

Coprime integers

Arithmetic

Journal of Combinatorial Theory Series A

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q-supercongruence with two parameters modulo [n]Φn(q)3. Here [n]=(1−qn)/(1−q) and Φn(q) is the n-th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q-analogue of Long's supercongruence. The latter is also a generalization of a recent q-supercongruence obtained by Guo and Schlosser.

Some q-supercongruences modulo the fourth power of a cyclotomic polynomial