Some q-Analogues of Granville and Sun’s Congruences
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Authors: W-W. QI
DOI: 10.46793/KgJMat2604.529Q
Abstract:
In this paper, we use q-binomial theorem to establish some new q-analogues of Granville and Sun’s congruence:
∑
k=1p−1 ≡ (mod p), |
and
∑
k=1p−1 ≡  (mod p), |
where x is a variable and p is an odd prime.
Keywords:
q-analogue, q-binomial theorem, cyclotomic polynomials.
References:
[1] A. Granville, The square of the Fermat quotient, Integers: Electrionic Journal of Combinatorial Number Theory 4(A22) (2004), 1–3.
[2] G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
[3] H. Pan, A q-analogue of Lehmer’s congruence, Acta Arith. 128 (2007), 303–318.
[4] L.-L. Shi and H. Pan, A q-analogue of Wolstenholme’s harmonic series congruence, Amer. Math. Monthly 6(114) (2007), 529–531. https://doi.org/10.1080/00029890.2007.11920441
[5] Z.-W. Sun, Congruences involing Bernoulli and Euler numbers, J. Number Theory 2(128) (2008), 280–312. https://doi.org/10.1016/j.jnt.2007.03.003
[6] R. Tauraso, Some q-analogue of congruences for center binomial sums, Colloq. Math. 133 (2013), 133–143. https://doi.org/10.4064/cm133-1-9
[7] L. Elkhiri, N. Omur and S. Kopal, A q-analogue of geanville’s congruence, Integers 22(A72) (2022).
[8] B. He, On q-congruences involving harmonic numbers, Ukrainian Mathematical Journal 69(9) (2018). https://doi.org/10.1007/s11253-081-1445-8
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