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A supercongruence motivated by the Legendre family of elliptic curves. (English) Zbl 1252.11017
Math. Notes 88, No. 4, 599-602 (2010); reprinted from Mat. Zametki 88, No. 4, 620-624 (2010).
The so-called supercongruences are usually proved through methods like the WZ algorithm, \(e.g.\), by W. Zudilin [J. Number Theory 129, No. 8, 1848–1857 (2009; Zbl 1231.11147)], or the Gaussian hypergeometric series, \(e.g.\), by S. Ahlgren and K. Ono [J. Reine Angew. Math. 518, 187–212 (2000; Zbl 0940.33002)] and by R. Osburn and C. Schneider [Math. Comput. 78, No. 265, 275–292 (2009; Zbl 1209.11049)], or the hypergeometric series evaluation identities, \(e.g.\), by D. McCarthy and R. Osburn [Arch. Math. 91, No. 6, 492–504 (2008; Zbl 1175.33004)].
Not so in this paper, where the following supercongruence for primes \(p>2\): \[ {\sum_{k=0}^{(p-1)/2}}\;2^{-3k}{\binom{2k}k}^2 \equiv\;(-1)^{(p-1)/2} {\sum_{k=0}^{(p-1)/2}}\;2^k {\binom{(p-1)/2}k}^2 \pmod{p^2}, \] is established using only elementary binomial identities and the Legendre transforms of certain sequences. The basis for such original proof is the following result: \[ {\sum_{k=0}^{(p-1)/2}}\;2^{-3k}{\binom{2k}k}^2 \equiv\;(-1)^{(p-1)/2} {\sum_{k=0}^{(p-1)/2}}\;2^k {\binom{(p-1)/2}k}^2 \pmod{p}, \] obtained by combining previous works from J. H. Silverman [The arithmetic of elliptic curves. New York etc.: Springer-Verlag (1986; Zbl 0585.14026)] and from C. H. Clemens [A scrapbook of complex curve theory. New York etc.: Plenum Press (1980; Zbl 0456.14016)].
The authors are also able to prove in an elegant way the following supercongruence: \[ {\sum_{k=0}^{(p-1)/2}}\;2^{-4k}{\binom{2k}k}^2 \equiv\;(-1)^{(p-1)/2} \pmod{p^2}, \] due to E. Mortenson [J. Number Theory 99, No. 1, 139–147 (2003; Zbl 1074.11045)] and they suggest the possibility to find similar elementary proofs for other supercongruences still provided by E. Mortenson [Trans. Am. Math. Soc. 355, No. 3, 987–1007 (2003; Zbl 1074.11044)].

MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
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References:
[1] J. H. Silverman, The Arithmetic of Elliptic Curves, in Grad. Texts in Math. (Springer-Verlag, New York, 1986), Vol. 106.
[2] C. H. Clemens, A Scrapbook of Complex Curve Theory, in Univ. Ser. Math. (Plenum Press, New York, 1980). · Zbl 0456.14016
[3] R. Osburn and C. Schneider, ”Gaussian hypergeometric series and supercongruences,” Math. Comp. 78(265), 275–292 (2009). · Zbl 1209.11049
[4] W. Zudilin, ”Ramanujan-type supercongruences,” J. Number Theory 129(8), 1848–1857 (2009). · Zbl 1231.11147
[5] S. Ahlgren and K. Ono, ”A Gaussian hypergeometric series evaluation and ApĂ©ery number congruences,” J. Reine Angew. Math. 518, 187–212 (2000). · Zbl 0940.33002
[6] D. McCarthy and R. Osburn, ”A p-adic analogue of a formula of Ramanujan,” Arch. Math. (Basel) 91(6), 492–504 (2008). · Zbl 1175.33004
[7] E. Mortenson, ”A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function,” J. Number Theory 99(1), 139–147 (2003). · Zbl 1074.11045
[8] E. Mortenson, ”Supercongruences between truncated 2 F 1 hypergeometric functions and their Gaussian analogs,” Trans. Amer.Math. Soc. 355(3), 987–1007 (2003). · Zbl 1074.11044
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