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Keywords:
congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number
congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number
Summary:
A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \pmod {p^4}$. For such a prime $p$, we establish an expression for ${2p-1\choose p-1}\pmod {p^8}$ given in terms of the sums $R_i:=\sum _{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$ \left ({2p-1\atop p-1}\right ) \equiv 1 -2p \sum _{k=1}^{p-1}\frac {1}{k} -2p^2\sum _{k=1}^{p-1}\frac {1}{k^2}\pmod {p^7}. $$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique of Helou and Terjanian, the above congruence is given as an expression involving the Bernoulli numbers.
References:
[1] Bayat, M.: A generalization of Wolstenholme's theorem. Am. Math. Mon. 104 (1997), 557-560. DOI 10.2307/2975083 | MR 1453658 | Zbl 0916.11002
[2] Crandall, R., Dilcher, K., Pomerance, C.: A search for Wieferich and Wilson primes. Math. Comput. 66 (1997), 433-449. DOI 10.1090/S0025-5718-97-00791-6 | MR 1372002 | Zbl 0854.11002
[3] Dilcher, K., Skula, L.: A new criterion for the first case of Fermat's last theorem. Math. Comp. 64 (1995), 363-392. MR 1248969 | Zbl 0817.11022
[4] Dilcher, K., Skula, L., Slavutsky, I. Sh.: Bernoulli Numbers. Bibliography (1713-1990). Queen's papers in Pure and Applied Mathematics 87 Queen's University, Kingston (1991), updated on-line version: www.mathstat.dal.ca/ {dilcher/bernoulli.html}. MR 1119305
[5] Glaisher, J. W. L.: Congruences relating to the sums of products of the first $n$ numbers and to other sums of products. Quart. J. 31 (1900), 1-35.
[6] Glaisher, J. W. L.: On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$. Quart. J. 31 (1900), 321-353.
[7] Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. J. Borwein, et al. Organic Mathematics Proc. of the workshop. Burnaby, 1995. CMS Conf. Proc. 20, American Mathematical Society, Providence (1997), 253-276. MR 1483922 | Zbl 0903.11005
[8] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. Clarendon Press Oxford (1979). MR 0568909 | Zbl 0423.10001
[9] Helou, C., Terjanian, G.: On Wolstenholme's theorem and its converse. J. Number Theory 128 (2008), 475-499. DOI 10.1016/j.jnt.2007.06.008 | MR 2389852 | Zbl 1236.11003
[10] Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84 Springer, New York (1982). MR 0661047 | Zbl 0482.10001
[11] Jacobson, N.: Basic Algebra. I. W. H. Freeman and Company New York (1985). MR 0780184 | Zbl 0557.16001
[12] Jakubec, S.: Note on the congruences $2^{p-1}\equiv 1\pmod {p^2}$, $3^{p-1}\equiv 1\pmod {p^2}$, $5^{p-1}\equiv 1\pmod {p^2}$. Acta Math. Inform. Univ. Ostrav. 6 (1998), 115-120. MR 1822520 | Zbl 1024.11002
[13] Jakubec, S.: Note on Wieferich's congruence for primes $p\equiv 1\pmod{4 }$. Abh. Math. Semin. Univ. Hamb. 68 (1998), 193-197. DOI 10.1007/BF02942562 | MR 1658393 | Zbl 0954.11009
[14] Kummer, E. E.: Über eine allgemeine Eigenschaft der rationalen Entwicklungscoëfficienten einer bestimmten Gattung analytischer Functionen. J. Reine Angew. Math. 41 (1851), 368-372 German.
[15] Lehmer, E.: On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Ann. Math. (2) 39 (1938), 350-360. MR 1503412 | Zbl 0019.00505
[16] McIntosh, R. J.: On the converse of Wolstenholme's theorem. Acta Arith. 71 (1995), 381-389. DOI 10.4064/aa-71-4-381-389 | MR 1339137 | Zbl 0829.11003
[17] McIntosh, R. J., Roettger, E. L.: A search for Fibonacci-Wieferich and Wolstenholme primes. Math. Comput. 76 (2007), 2087-2094. DOI 10.1090/S0025-5718-07-01955-2 | MR 2336284 | Zbl 1139.11003
[18] Meštrović, R.: On the mod $p^7$ determination of ${2p-1\choose p-1}$. Rocky Mt. J. Math. 44 (2014), 633-648; preprint arXiv:1108.1174v1 [math.NT] (2011) . MR 3240517
[19] Meštrović, R.: Wolstenholme's theorem: its generalizations and extensions in the last hundred and fifty years (1862-2012). preprint arXiv:1111.3057v2 [math.NT] (2011).
[20] Meštrović, R.: Some Wolstenholme type congruences. Math. Appl., Brno 2 (2013), 35-42. DOI 10.13164/ma.2013.04 | MR 3275598
[21] Ribenboim, P.: 13 Lectures on Fermat's Last Theorem. Springer New York (1979). MR 0551363 | Zbl 0456.10006
[22] Skula, L.: Fermat's last theorem and the Fermat quotients. Comment. Math. Univ. St. Pauli 41 (1992), 35-54. MR 1166223 | Zbl 0753.11016
[23] Wolstenholme, J.: On certain properties of prime numbers. Quart. J. Pure Appl. Math. 5 (1862), 35-39.
[24] Zhao, J.: Wolstenholme type theorem for multiple harmonic sums. Int. J. Number Theory 4 (2008), 73-106. DOI 10.1142/S1793042108001146 | MR 2387917 | Zbl 1218.11005
[25] Zhao, J.: Bernoulli numbers, Wolstenholme's theorem, and $p^5$ variations of Lucas' theorem. J. Number Theory 123 (2007), 18-26. DOI 10.1016/j.jnt.2006.05.005 | MR 2295427
[26] Zhou, X., Cai, T.: A generalization of a curious congruence on harmonic sums. Proc. Am. Math. Soc. 135 (2007), 1329-1333. DOI 10.1090/S0002-9939-06-08777-6 | MR 2276641 | Zbl 1115.11006
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