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Keywords:
congruence; Fermat quotient; harmonic numbers
congruence; Fermat quotient; harmonic numbers
Summary:
Let $p\ge 5$ be a prime, and let $q_p(2):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. The following curious congruence was conjectured by L. Skula and proved by A. Granville
\[ q_p(2)^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}\quad(\operatorname{mod} p)\,. \] In this note we establish the above congruence by entirely elementary number theory arguments.
References:
[1] Agoh, T., Dilcher, K., Skula, L.: Fermat quotients for composite moduli. J. Number Theory 66 (1997), 29–50. DOI 10.1006/jnth.1997.2162 | MR 1467188 | Zbl 0884.11003
[2] Cao, H. Q., Pan, H.: A congruence involving product of $q$–binomial coefficients. J. Number Theory 121 (2006), 224–233. DOI 10.1016/j.jnt.2006.02.004 | MR 2274904
[3] Ernvall, R., Metsänkylä, T.: On the $p$–divisibility of Fermat quotients. Math. Comp. 66 (1997), 1353–1365. DOI 10.1090/S0025-5718-97-00843-0 | MR 1408373 | Zbl 0903.11002
[4] Glaisher, J. W. L.: On the residues of the sums of the inverse powers of numbers in arithmetical progression. Quart. J. Math. 32 (1900), 271–288.
[5] Granville, A.: Some conjectures related to Fermat’s Last Theorem. Number Theory (Banff, AB, 1988), de Gruyter, Berlin (1990), 177–192. MR 1106660 | Zbl 0702.11015
[6] Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. Organic Mathematics–Burnaby, BC 1995, CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. MR 1483922 | Zbl 0903.11005
[7] Granville, A.: The square of the Fermat quotient. Integers 4 (2004), # A22. MR 2116007 | Zbl 1083.11005
[8] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. Clarendon Press, Oxford, 1960. Zbl 0086.25803
[9] Lehmer, E.: On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Ann. Math. (1938), 350–360. MR 1503412 | Zbl 0019.00505
[10] Ribenboim, P.: 13 lectures on Fermat’s last theorem. Springer–Verlag, New York, Heidelberg, Berlin, 1979. MR 0551363 | Zbl 0456.10006
[11] Slavutsky, I. Sh.: Leudesdorf’s theorem and Bernoulli numbers. Arch. Math. 35 (1999), 299–303.
[12] Sun, Z. H.: Congruences concerning Bernoulli numbers and Bernoulli polynomials. Discrete Appl. Math. 105 (1–3) (2000), 193–223. DOI 10.1016/S0166-218X(00)00184-0 | MR 1780472 | Zbl 0990.11008
[13] Sun, Z. W.: Arithmetic theory of harmonic numbers. Proc. Amer. Math. Soc. 140 (2012), 415–428. DOI 10.1090/S0002-9939-2011-10925-0 | MR 2846311
[14] Tauraso, R.: Congruences involving alternating multiple harmonic sums. Electron. J. Comb. 17 (2010), # R16. MR 2587747 | Zbl 1222.11006
[15] Wolstenholme, J.: On certain properties of prime numbers. Quart. J. Pure Appl. Math. 5 (1862), 35–39.
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