Congruences for Wolstenholme primes

Romeo Meštrović

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 237-253
  • ISSN: 0011-4642

Abstract

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A prime p is said to be a Wolstenholme prime if it satisfies the congruence 2 p - 1 p - 1 ≡ 1 ( mod p 4 ) . For such a prime p , we establish an expression for 2 p - 1 p - 1 ( mod p 8 ) given in terms of the sums R i : = ∑ 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 2 p - 1 p - 1 ≡ 1 - 2 p ∑ k = 1 p - 1 1 k - 2 p 2 ∑ k = 1 p - 1 1 k 2 ( mod 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.

How to cite

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Meštrović, Romeo. "Congruences for Wolstenholme primes." Czechoslovak Mathematical Journal 65.1 (2015): 237-253. <http://eudml.org/doc/270050>.

@article{Meštrović2015,
abstract = {A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $\{2p-1\atopwithdelims ()p-1\} \equiv 1 \hspace\{4.44443pt\}(\@mod \; p^4)$. For such a prime $p$, we establish an expression for $\{2p-1\atopwithdelims ()p-1\}\hspace\{4.44443pt\}(\@mod \; p^8)$ 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 \[ \left(\{2p-1\atop p-1\}\right) \equiv 1 -2p \sum \_\{k=1\}^\{p-1\}\frac\{1\}\{k\} -2p^2\sum \_\{k=1\}^\{p-1\}\frac\{1\}\{k^2\}\hspace\{10.0pt\}(\@mod \; p^7). \] 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.},
author = {Meštrović, Romeo},
journal = {Czechoslovak Mathematical Journal},
keywords = {congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number; congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number},
language = {eng},
number = {1},
pages = {237-253},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Congruences for Wolstenholme primes},
url = {http://eudml.org/doc/270050},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Meštrović, Romeo
TI - Congruences for Wolstenholme primes
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 237
EP - 253
AB - A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\atopwithdelims ()p-1} \equiv 1 \hspace{4.44443pt}(\@mod \; p^4)$. For such a prime $p$, we establish an expression for ${2p-1\atopwithdelims ()p-1}\hspace{4.44443pt}(\@mod \; p^8)$ 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 \[ \left({2p-1\atop p-1}\right) \equiv 1 -2p \sum _{k=1}^{p-1}\frac{1}{k} -2p^2\sum _{k=1}^{p-1}\frac{1}{k^2}\hspace{10.0pt}(\@mod \; p^7). \] 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.
LA - eng
KW - congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number; congruence; prime power; Wolstenholme prime; Wolstenholme's theorem; Bernoulli number
UR - http://eudml.org/doc/270050
ER -

References

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  1. Bayat, M., 10.2307/2975083, Am. Math. Mon. 104 (1997), 557-560. (1997) Zbl0916.11002MR1453658DOI10.2307/2975083
  2. Crandall, R., Dilcher, K., Pomerance, C., 10.1090/S0025-5718-97-00791-6, Math. Comput. 66 (1997), 433-449. (1997) Zbl0854.11002MR1372002DOI10.1090/S0025-5718-97-00791-6
  3. Dilcher, K., Skula, L., A new criterion for the first case of Fermat's last theorem, Math. Comp. 64 (1995), 363-392. (1995) Zbl0817.11022MR1248969
  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}. (1991) MR1119305
  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. (1900) 
  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. (1900) 
  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. (1997) Zbl0903.11005MR1483922
  8. Hardy, G. H., Wright, E. M., An Introduction to the Theory of Numbers, Clarendon Press Oxford (1979). (1979) Zbl0423.10001MR0568909
  9. Helou, C., Terjanian, G., 10.1016/j.jnt.2007.06.008, J. Number Theory 128 (2008), 475-499. (2008) Zbl1236.11003MR2389852DOI10.1016/j.jnt.2007.06.008
  10. Ireland, K., Rosen, M., A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84 Springer, New York (1982). (1982) Zbl0482.10001MR0661047
  11. Jacobson, N., Basic Algebra. I, W. H. Freeman and Company New York (1985). (1985) Zbl0557.16001MR0780184
  12. Jakubec, S., Note on the congruences 2 p - 1 ≡ 1 ( mod p 2 ) , 3 p - 1 ≡ 1 ( mod p 2 ) , 5 p - 1 ≡ 1 ( mod p 2 ) , Acta Math. Inform. Univ. Ostrav. 6 (1998), 115-120. (1998) Zbl1024.11002MR1822520
  13. Jakubec, S., 10.1007/BF02942562, Abh. Math. Semin. Univ. Hamb. 68 (1998), 193-197. (1998) Zbl0954.11009MR1658393DOI10.1007/BF02942562
  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. (1851) 
  15. Lehmer, E., On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. (2) 39 (1938), 350-360. (1938) Zbl0019.00505MR1503412
  16. McIntosh, R. J., 10.4064/aa-71-4-381-389, Acta Arith. 71 (1995), 381-389. (1995) Zbl0829.11003MR1339137DOI10.4064/aa-71-4-381-389
  17. McIntosh, R. J., Roettger, E. L., 10.1090/S0025-5718-07-01955-2, Math. Comput. 76 (2007), 2087-2094. (2007) Zbl1139.11003MR2336284DOI10.1090/S0025-5718-07-01955-2
  18. Meštrović, R., On the mod p 7 determination of 2 p - 1 p - 1 , Rocky Mt. J. Math. 44 (2014), 633-648; preprint arXiv:1108.1174v1 [math.NT] (2011) . (2014) MR3240517
  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). (2011) 
  20. Meštrović, R., 10.13164/ma.2013.04, Math. Appl., Brno 2 (2013), 35-42. (2013) MR3275598DOI10.13164/ma.2013.04
  21. Ribenboim, P., 13 Lectures on Fermat's Last Theorem, Springer New York (1979). (1979) Zbl0456.10006MR0551363
  22. Skula, L., Fermat's last theorem and the Fermat quotients, Comment. Math. Univ. St. Pauli 41 (1992), 35-54. (1992) Zbl0753.11016MR1166223
  23. Wolstenholme, J., On certain properties of prime numbers, Quart. J. Pure Appl. Math. 5 (1862), 35-39. (1862) 
  24. Zhao, J., 10.1142/S1793042108001146, Int. J. Number Theory 4 (2008), 73-106. (2008) Zbl1218.11005MR2387917DOI10.1142/S1793042108001146
  25. Zhao, J., 10.1016/j.jnt.2006.05.005, J. Number Theory 123 (2007), 18-26. (2007) MR2295427DOI10.1016/j.jnt.2006.05.005
  26. Zhou, X., Cai, T., 10.1090/S0002-9939-06-08777-6, Proc. Am. Math. Soc. 135 (2007), 1329-1333. (2007) Zbl1115.11006MR2276641DOI10.1090/S0002-9939-06-08777-6

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