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Congruences concerning Lucas sequences. (English) Zbl 1294.11003

Motivated by an algebraic investigation of the author’s brother Z. Sun [“Various congruences involving binomial coefficients and higher-order Catalan numbers”, Preprint, arXiv:0909.3808], this paper presents new congruences for \(\sum_{k=0}^{[p/4]}{4k\choose 2k}/m^k\) and \(\sum_{k=0}^{[p/3]}{3k\choose k}/m^k\) modulo \(p\), where \(p>3\) is prime and \(m\) is a rational \(p\)-adic integer with \(m \not\equiv 0\pmod p\).
The methods applied go beyond cubic residues and congruences, reduced and binary quadratic forms, expansions and congruences for Lucas sequences (available, e.g., in [H. C. Williams, Édouard Lucas and primality testing. New York, NY: John Wiley & Sons (1998; Zbl 1155.11363)]). In the proof the author employs a congruence found by D. H. Lehmer [Ann. Math. (2) 31, 419–448 (1930; JFM 56.0874.04)] and theorems from the author and his brother [Acta Arith. 60, No. 4, 371–388 (1992; Zbl 0725.11009)] and from the author [J. Nanjing Univ., Math. Biq. 10, No. 1, 105–118 (1993; Zbl 0808.11016); ibid. 12, No. 1, 90–102 (1995; Zbl 0862.11023); Rocky Mt. J. Math. 33, No. 3, 1123–1145 (2003; Zbl 1076.11009); J. Number Theory 102, No. 1, 41–89 (2003; Zbl 1033.11003); ibid. 128, No. 5, 1295–1335 (2008; Zbl 1137.11003); Proc. Am. Math. Soc. 139, No. 6, 1915–1929 (2011; Zbl 1225.11006); J. Number Theory 133, No. 6, 1950–1976 (2013; Zbl 1277.11002)].
The paper is related also to some other works (e.g., by Z.-W. Sun and R. Tauraso [Adv. Appl. Math. 45, No. 1, 125–148 (2010; Zbl 1231.11021)], by L. Zhao, H. Pan and Z. Sun [Proc. Am. Math. Soc. 138, No. 1, 37–46 (2010; Zbl 1219.11034)] and by the author [J. Number Theory 133, No. 5, 1572–1595 (2013; Zbl 1300.11007)]) all inspired by an open conjecture due to A. Adamchuk [“Comments on OEIS A066796 in 2006”, http://oeis.org/A066796].

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A15 Power residues, reciprocity
11E25 Sums of squares and representations by other particular quadratic forms
05A10 Factorials, binomial coefficients, combinatorial functions

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References:

[1] DOI: 10.2307/1968235 · JFM 56.0874.04 · doi:10.2307/1968235
[2] Sun Z. H., J. Nanjing Univ. Math. Biquarterly 9 pp 227–
[3] Sun Z. H., J. Nanjing Univ. Math. Biquarterly 10 pp 105–
[4] Sun Z. H., J. Nanjing Univ. Math. Biquarterly 12 pp 90–
[5] Sun Z. H., Acta Arith. 84 pp 291–
[6] DOI: 10.1216/rmjm/1181069947 · Zbl 1076.11009 · doi:10.1216/rmjm/1181069947
[7] DOI: 10.1016/S0022-314X(03)00067-2 · Zbl 1033.11003 · doi:10.1016/S0022-314X(03)00067-2
[8] DOI: 10.1016/j.jnt.2006.08.001 · Zbl 1124.11006 · doi:10.1016/j.jnt.2006.08.001
[9] DOI: 10.1016/j.jnt.2007.07.009 · Zbl 1137.11003 · doi:10.1016/j.jnt.2007.07.009
[10] DOI: 10.1090/S0002-9939-2010-10566-X · Zbl 1225.11006 · doi:10.1090/S0002-9939-2010-10566-X
[11] DOI: 10.1016/j.jnt.2012.10.001 · Zbl 1300.11007 · doi:10.1016/j.jnt.2012.10.001
[12] DOI: 10.1016/j.jnt.2012.11.004 · Zbl 1277.11002 · doi:10.1016/j.jnt.2012.11.004
[13] Sun Z. H., Acta Arith. 60 pp 371–
[14] DOI: 10.1016/j.aam.2010.01.001 · Zbl 1231.11021 · doi:10.1016/j.aam.2010.01.001
[15] H. C. Williams, Édouard Lucas and Primality Testing, Canadian Mathematical Society Series of Monographs and Advanced Texts 22 (Wiley, New York, 1998) pp. 74–92.
[16] DOI: 10.1090/S0002-9939-09-10067-9 · Zbl 1219.11034 · doi:10.1090/S0002-9939-09-10067-9
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