# Congruences involving the Fermat quotient

• Volume: 63, Issue: 4, page 949-968
• ISSN: 0011-4642

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## Abstract

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Let $p>3$ be a prime, and let ${q}_{p}\left(2\right)=\left({2}^{p-1}-1\right)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $\sum _{k=1}^{p-1}\frac{1}{k·{2}^{k}}\equiv {q}_{p}\left(2\right)-\frac{p{q}_{p}{\left(2\right)}^{2}}{2}+\frac{{p}^{2}{q}_{p}{\left(2\right)}^{3}}{3}-\frac{7}{48}{p}^{2}{B}_{p-3}\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{3}\right),$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that ${q}_{p}{\left(2\right)}^{3}\equiv -3\sum _{k=1}^{p-1}\frac{{2}^{k}}{{k}^{3}}+\frac{7}{16}\sum _{k=1}^{\left(p-1\right)/2}\frac{1}{{k}^{3}}\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}p\right),$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ${\sum }_{k=1}^{p-1}1/\left({k}^{2}·{2}^{k}\right)$ modulo ${p}^{2}$ that also generalizes a related Sun’s congruence modulo $p$.

## How to cite

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Meštrović, Romeo. "Congruences involving the Fermat quotient." Czechoslovak Mathematical Journal 63.4 (2013): 949-968. <http://eudml.org/doc/260793>.

@article{Meštrović2013,
abstract = {Let $p>3$ be a prime, and let $q_p(2)=(2^\{p-1\}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $\sum \_\{k=1\}^\{p-1\} \frac\{1\}\{k\cdot 2^k\} \equiv q\_p(2)-\frac\{pq\_p(2)^2\}\{2\}+ \frac\{p^2 q\_p(2)^3\}\{3\} -\frac\{7\}\{48\} p^2 B\_\{p-3\}\hspace\{10.0pt\}(\@mod \; p^3),$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $q\_p(2)^3 \equiv -3\sum \_\{k=1\}^\{p-1\} \frac\{2^k\}\{k^3\}+ \frac\{7\}\{16\} \sum \_\{k=1\}^\{(p-1)/2\} \frac\{1\}\{k^3\} \hspace\{10.0pt\}(\@mod \; p),$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum _\{k=1\}^\{p-1\}1/(k^2\cdot 2^k)$ modulo $p^2$ that also generalizes a related Sun’s congruence modulo $p$.},
author = {Meštrović, Romeo},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fermat quotient; $n$th harmonic number of order $m$; Bernoulli number; Fermat quotient; Bernoulli number},
language = {eng},
number = {4},
pages = {949-968},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Congruences involving the Fermat quotient},
url = {http://eudml.org/doc/260793},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Meštrović, Romeo
TI - Congruences involving the Fermat quotient
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 949
EP - 968
AB - Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $\sum _{k=1}^{p-1} \frac{1}{k\cdot 2^k} \equiv q_p(2)-\frac{pq_p(2)^2}{2}+ \frac{p^2 q_p(2)^3}{3} -\frac{7}{48} p^2 B_{p-3}\hspace{10.0pt}(\@mod \; p^3),$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $q_p(2)^3 \equiv -3\sum _{k=1}^{p-1} \frac{2^k}{k^3}+ \frac{7}{16} \sum _{k=1}^{(p-1)/2} \frac{1}{k^3} \hspace{10.0pt}(\@mod \; p),$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum _{k=1}^{p-1}1/(k^2\cdot 2^k)$ modulo $p^2$ that also generalizes a related Sun’s congruence modulo $p$.
LA - eng
KW - Fermat quotient; $n$th harmonic number of order $m$; Bernoulli number; Fermat quotient; Bernoulli number
UR - http://eudml.org/doc/260793
ER -

## References

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1. Agoh, T., Dilcher, K., Skula, L., 10.1006/jnth.1997.2162, J. Number Theory 66 (1997), 29-50. (1997) Zbl0884.11003MR1467188DOI10.1006/jnth.1997.2162
2. Agoh, T., Dilcher, K., Skula, L., 10.1090/S0025-5718-98-00951-X, Math. Comput. 67 (1998), 843-861. (1998) Zbl1024.11003MR1464140DOI10.1090/S0025-5718-98-00951-X
3. Agoh, T., Skula, L., 10.1016/j.jnt.2008.06.001, J. Number Theory 128 (2008), 2865-2873. (2008) Zbl1195.11009MR2457841DOI10.1016/j.jnt.2008.06.001
4. Cao, H.-Q., Pan, H., 10.1016/j.jnt.2006.02.004, J. Number Theory 121 (2006), 224-233. (2006) Zbl1135.11003MR2274904DOI10.1016/j.jnt.2006.02.004
5. Crandall, R., Dilcher, K., Pomerance, C., 10.1090/S0025-5718-97-00791-6, Math. Comp. 66 (1997), 433-449. (1997) Zbl0854.11002MR1372002DOI10.1090/S0025-5718-97-00791-6
6. Dilcher, K., Skula, L., A new criterion for the first case of Fermat's last theorem, Math. Comput. 64 (1995), 363-392. (1995) Zbl0817.11022MR1248969
7. Dilcher, K., Skula, L., The cube of the Fermat quotient, Integers (electronic only) 6 (2006), Paper A24, 12 pages. (2006) Zbl1103.11011MR2264839
8. Dilcher, K., Skula, L., Slavutskii, I. S., eds., Bernoulli numbers. Bibliography (1713-1990). Enlarged ed. Queen's Papers in Pure and Applied Mathematics 87, Queen's University Kingston (1991). (1991) MR1119305
9. Dobson, J. B., On Lerch's formula for the Fermat quotient, Preprint, arXiv:1103.3907v3, 2012.
10. Dorais, F. G., Klyve, D., A Wieferich prime search up to $6·7×{10}^{15}$, J. Integer Seq. (electronic only) 14 (2011), Article 11.9.2, 14 pages. (2011) MR2859986
11. Eisenstein, G., Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhängen und durch gewisse lineare Funktional-Gleichungen definiert werden, Bericht. K. Preuss. Akad. Wiss. Berlin 15 (1850), 36-42 <title>Mathematische Werke. Band II Chelsea Publishing Company New York 705-711 (1975), German. (1975)
12. Ernvall, R., Metsänkylä, T., 10.1090/S0025-5718-97-00843-0, Math. Comput. 66 (1997), 1353-1365. (1997) Zbl0903.11002MR1408373DOI10.1090/S0025-5718-97-00843-0
13. 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)
14. Glaisher, J. W. L., On the residues of ${r}^{p-1}$ to modulus ${p}^{2}$, ${p}^{3}$, etc, Quart. J. 32 (1900), 1-27. (1900)
15. Granville, A., Arithmetic properties of binomial coefficients. I: Binomial coefficients modulo prime powers, Organic Mathematics. Proceedings of the workshop, Simon Fraser University, Burnaby, Canada, December 12-14, 1995. CMS Conf. Proc. 20 J. Borwein et al. American Mathematical Society Providence (1997), 253-276. (1997) Zbl0903.11005MR1483922
16. Granville, A., Some conjectures related to Fermat's Last Theorem, Number Theory. Proceedings of the first conference of the Canadian Number Theory Association held at the Banff Center, Banff, Alberta, Canada, April 17-27, 1988 R. A. Mollin Walter de Gruyter Berlin (1990), 177-192. (1990) Zbl0702.11015MR1106660
17. Granville, A., The square of the Fermat quotient, Integers 4 (2004), Paper A22, 3 pages, electronic only. (2004) Zbl1083.11005MR2116007
18. Jakubec, S., Note on the congruences ${2}^{p-1}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{2}\right)$, ${3}^{p-1}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{2}\right)$, ${5}^{p-1}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{2}\right)$, Acta Math. Inform. Univ. Ostrav. 6 (1998), 115-120. (1998) Zbl1024.11002MR1822520
19. Jakubec, S., 10.1007/BF02942562, Abh. Math. Semin. Univ. Hamb. 68 (1998), 193-197. (1998) Zbl0954.11009MR1658393DOI10.1007/BF02942562
20. Jakubec, S., 10.2478/s12175-007-0052-1, Math. Slovaca 58 (2008), 19-30. (2008) Zbl1164.11014MR2372823DOI10.2478/s12175-007-0052-1
21. Kohnen, W., 10.2307/2974738, Am. Math. Mon. 104 (1997), 444-445. (1997) Zbl0880.11002MR1447978DOI10.2307/2974738
22. Lehmer, E., On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360. (1938) Zbl0019.00505MR1503412
23. Lerch, M., 10.1007/BF01561092, Math. Ann. 60 (1905), 471-490 German. (1905) MR1511321DOI10.1007/BF01561092
24. Meštrović, R., 10.1017/S0004972711002826, Bull. Aust. Math. Soc. 85 (2012), 482-496. (2012) Zbl1273.11033MR2924776DOI10.1017/S0004972711002826
25. Meštrović, R., 10.5817/AM2012-2-113, Arch. Math., Brno 48 (2012), 113-120. (2012) Zbl1259.11006MR2946211DOI10.5817/AM2012-2-113
26. Pan, H., On a generalization of Carlitz's congruence, Int. J. Mod. Math. 4 (2009), 87-93. (2009) Zbl1247.11025MR2508944
27. Ribenboim, P., 13 Lectures on Fermat's Last Theorem, Springer New York (1979). (1979) Zbl0456.10006MR0551363
28. Skula, L., A Remark on Mirimanoff polynomials, Comment. Math. Univ. St. Pauli 31 (1982), 89-97. (1982) Zbl0496.10006MR0674586
29. Skula, L., Fermat and Wilson quotients for $p$-adic integers, Acta Math. Inform. Univ. Ostrav. 6 (1998), 167-181. (1998) Zbl1025.11001MR1822528
30. Skula, L., Fermat's Last theorem and the Fermat quotients, Comment. Math. Univ. St. Pauli 41 (1992), 35-54. (1992) Zbl0753.11016MR1166223
31. Skula, L., 10.2478/s12175-007-0050-3, Math. Slovaca 58 (2008), 5-10. (2008) Zbl1164.11001MR2372821DOI10.2478/s12175-007-0050-3
32. Slavutsky, I. S., Leudesdorf's theorem and Bernoulli numbers, Arch. Math., Brno 35 (1999), 299-303. (1999) MR1744517
33. Spivey, M. Z., 10.1016/j.disc.2007.03.052, Discrete Math. 307 (2007), 3130-3146. (2007) Zbl1129.05006MR2370116DOI10.1016/j.disc.2007.03.052
34. Sun, Z. H., 10.1016/S0166-218X(00)00184-0, Discrete Appl. Math. 105 (2000), 193-223. (2000) Zbl0990.11008MR1780472DOI10.1016/S0166-218X(00)00184-0
35. Sun, Z. H., 10.1016/j.jnt.2007.03.003, J. Number Theory 128 (2008), 280-312. (2008) Zbl1154.11010MR2380322DOI10.1016/j.jnt.2007.03.003
36. Sun, Z. H., Five congruences for primes, Fibonacci Q. 40 (2002), 345-351. (2002) Zbl1009.11004MR1920576
37. Sun, Z. H., The combinatorial sum ${\sum }_{k=0,k\equiv r\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its applications in number theory II, J. Nanjing Univ., Math. Biq. 10 (1993), 105-118 Chinese. (1993) MR1248315
38. Sun, Z. W., 10.1090/S0002-9939-1995-1242105-X, Proc. Am. Math. Soc. 123 (1995), 1341-1346. (1995) Zbl0833.11001MR1242105DOI10.1090/S0002-9939-1995-1242105-X
39. Sun, Z. W., 10.1007/s11425-010-3151-3, Sci. China, Math. 53 (2010), 2473-2488. (2010) Zbl1221.11054MR2718841DOI10.1007/s11425-010-3151-3
40. Sun, Z. W., 10.1007/BF02785421, Isr. J. Math. 128 (2002), 135-156. (2002) MR1910378DOI10.1007/BF02785421
41. Sylvester, J. J., Sur une propriété des nombres premiers qui se ratache au théoreme de Fermat, C. R. Acad. Sci. Paris 52 (1861), 161-163 <title>The Collected Mathematical Papers of James Joseph Sylvester. Volume II (1854-1873). With Two Plates Cambridge University Press Cambridge 229-231 (1908). (1908)
42. Tauraso, R., Congruences involving alternating multiple harmonic sums, Electron. J. Comb. 17 (2010), Research Paper R16, 11 pages. (2010) Zbl1222.11006MR2587747
43. Wieferich, A., On Fermat's Last Theorem, J. für Math. 136 (1909), 293-302 German. (1909)
44. Wolstenholme, J., On certain properties of prime numbers, Quart. J. 5 (1862), 35-39. (1862)

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