# Article

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Keywords:
Fermat quotient; $n$th harmonic number of order $m$; Bernoulli number
Summary:
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 $$\sum _{k=1}^{p-1} \frac {1}{k\cdot 2^k} \equiv q_p(2)-\frac {pq_p(2)^2}{2}+ \frac {p^2 q_p(2)^3}{3} -\frac {7}{48} p^2 B_{p-3}\pmod {p^3},$$ 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 $$q_p(2)^3 \equiv -3\sum _{k=1}^{p-1} \frac {2^k}{k^3}+ \frac {7}{16} \sum _{k=1}^{(p-1)/2} \frac {1}{k^3} \pmod {p},$$ 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 $p$.
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] Agoh, T., Dilcher, K., Skula, L.: Wilson quotients for composite moduli. Math. Comput. 67 (1998), 843-861. DOI 10.1090/S0025-5718-98-00951-X | MR 1464140 | Zbl 1024.11003
[3] Agoh, T., Skula, L.: The fourth power of the Fermat quotient. J. Number Theory 128 (2008), 2865-2873. DOI 10.1016/j.jnt.2008.06.001 | MR 2457841 | Zbl 1195.11009
[4] Cao, H.-Q., Pan, H.: A congruence involving products of $q$-binomial coefficients. J. Number Theory 121 (2006), 224-233. DOI 10.1016/j.jnt.2006.02.004 | MR 2274904 | Zbl 1135.11003
[5] Crandall, R., Dilcher, K., Pomerance, C.: A search for Wieferich and Wilson primes. Math. Comp. 66 (1997), 433-449. DOI 10.1090/S0025-5718-97-00791-6 | MR 1372002 | Zbl 0854.11002
[6] Dilcher, K., Skula, L.: A new criterion for the first case of Fermat's last theorem. Math. Comput. 64 (1995), 363-392. MR 1248969 | Zbl 0817.11022
[7] Dilcher, K., Skula, L.: The cube of the Fermat quotient. Integers (electronic only) 6 (2006), Paper A24, 12 pages. MR 2264839 | Zbl 1103.11011
[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). MR 1119305
[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 $6\cdot 7\times 10^{15}$. J. Integer Seq. (electronic only) 14 (2011), Article 11.9.2, 14 pages. MR 2859986
[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 Mathematische Werke. Band II Chelsea Publishing Company New York 705-711 (1975), German.</div> <div class="reference">[12] Ernvall, R., Metsänkylä, T.: <b>On the $p$-divisibility of Fermat quotients</b>. Math. Comput. 66 (1997), 1353-1365. <a href="http://dx.doi.org/10.1090/S0025-5718-97-00843-0" target="_blank">DOI 10.1090/S0025-5718-97-00843-0</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1408373" target="_blank">MR 1408373</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0903.11002" target="_blank">Zbl 0903.11002</a></div> <div class="reference">[13] Glaisher, J. W. L.: <b>On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$</b>. Quart. J. 31 (1900), 321-353.</div> <div class="reference">[14] Glaisher, J. W. L.: <b>On the residues of $r^{p-1}$ to modulus $p^2$, $p^3$, etc</b>. Quart. J. 32 (1900), 1-27.</div> <div class="reference">[15] Granville, A.: <b>Arithmetic properties of binomial coefficients. I: Binomial coefficients modulo prime powers</b>. 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. <a href="http://www.ams.org/mathscinet-getitem?mr=1483922" target="_blank">MR 1483922</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0903.11005" target="_blank">Zbl 0903.11005</a></div> <div class="reference">[16] Granville, A.: <b>Some conjectures related to Fermat's Last Theorem</b>. 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. <a href="http://www.ams.org/mathscinet-getitem?mr=1106660" target="_blank">MR 1106660</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0702.11015" target="_blank">Zbl 0702.11015</a></div> <div class="reference">[17] Granville, A.: <b>The square of the Fermat quotient</b>. Integers 4 (2004), Paper A22, 3 pages, electronic only. <a href="http://www.ams.org/mathscinet-getitem?mr=2116007" target="_blank">MR 2116007</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1083.11005" target="_blank">Zbl 1083.11005</a></div> <div class="reference">[18] Jakubec, S.: <b>Note on the congruences $2^{p-1}\equiv 1 \pmod{p^2}$, $3^{p-1}\equiv 1 \pmod{p^2}$, $5^{p-1}\equiv 1 \pmod{p^2}$</b>. Acta Math. Inform. Univ. Ostrav. 6 (1998), 115-120. <a href="http://www.ams.org/mathscinet-getitem?mr=1822520" target="_blank">MR 1822520</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1024.11002" target="_blank">Zbl 1024.11002</a></div> <div class="reference">[19] Jakubec, S.: <b>Note on Wieferich's congruence for primes $p\equiv 1\pmod{4 }$</b>. Abh. Math. Semin. Univ. Hamb. 68 (1998), 193-197. <a href="http://dx.doi.org/10.1007/BF02942562" target="_blank">DOI 10.1007/BF02942562</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1658393" target="_blank">MR 1658393</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0954.11009" target="_blank">Zbl 0954.11009</a></div> <div class="reference">[20] Jakubec, S.: <b>Connection between Fermat quotients and Euler numbers</b>. Math. Slovaca 58 (2008), 19-30. <a href="http://dx.doi.org/10.2478/s12175-007-0052-1" target="_blank">DOI 10.2478/s12175-007-0052-1</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2372823" target="_blank">MR 2372823</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1164.11014" target="_blank">Zbl 1164.11014</a></div> <div class="reference">[21] Kohnen, W.: <b>A simple congruence modulo $p$</b>. Am. Math. Mon. 104 (1997), 444-445. <a href="http://dx.doi.org/10.2307/2974738" target="_blank">DOI 10.2307/2974738</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1447978" target="_blank">MR 1447978</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0880.11002" target="_blank">Zbl 0880.11002</a></div> <div class="reference">[22] Lehmer, E.: <b>On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson</b>. Ann. Math. 39 (1938), 350-360. <a href="http://www.ams.org/mathscinet-getitem?mr=1503412" target="_blank">MR 1503412</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0019.00505" target="_blank">Zbl 0019.00505</a></div> <div class="reference">[23] Lerch, M.: <b>Zur Theorie des Fermatschen Quotienten $(a^{p-1}-1)/p=q(a)$</b>. Math. Ann. 60 (1905), 471-490 German. <a href="http://dx.doi.org/10.1007/BF01561092" target="_blank">DOI 10.1007/BF01561092</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1511321" target="_blank">MR 1511321</a></div> <div class="reference">[24] Meštrović, R.: <b>An extension of Sury's identity and related congruences</b>. Bull. Aust. Math. Soc. 85 (2012), 482-496. <a href="http://dx.doi.org/10.1017/S0004972711002826" target="_blank">DOI 10.1017/S0004972711002826</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2924776" target="_blank">MR 2924776</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1273.11033" target="_blank">Zbl 1273.11033</a></div> <div class="reference">[25] Meštrović, R.: <b>An elementary proof of a congruence by Skula and Granville</b>. Arch. Math., Brno 48 (2012), 113-120. <a href="http://dx.doi.org/10.5817/AM2012-2-113" target="_blank">DOI 10.5817/AM2012-2-113</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2946211" target="_blank">MR 2946211</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1259.11006" target="_blank">Zbl 1259.11006</a></div> <div class="reference">[26] Pan, H.: <b>On a generalization of Carlitz's congruence</b>. Int. J. Mod. Math. 4 (2009), 87-93. <a href="http://www.ams.org/mathscinet-getitem?mr=2508944" target="_blank">MR 2508944</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1247.11025" target="_blank">Zbl 1247.11025</a></div> <div class="reference">[27] Ribenboim, P.: <b>13 Lectures on Fermat's Last Theorem</b>. Springer New York (1979). <a href="http://www.ams.org/mathscinet-getitem?mr=0551363" target="_blank">MR 0551363</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0456.10006" target="_blank">Zbl 0456.10006</a></div> <div class="reference">[28] Skula, L.: <b>A Remark on Mirimanoff polynomials</b>. Comment. Math. Univ. St. Pauli 31 (1982), 89-97. <a href="http://www.ams.org/mathscinet-getitem?mr=0674586" target="_blank">MR 0674586</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0496.10006" target="_blank">Zbl 0496.10006</a></div> <div class="reference">[29] Skula, L.: <b>Fermat and Wilson quotients for $p$-adic integers</b>. Acta Math. Inform. Univ. Ostrav. 6 (1998), 167-181. <a href="http://www.ams.org/mathscinet-getitem?mr=1822528" target="_blank">MR 1822528</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1025.11001" target="_blank">Zbl 1025.11001</a></div> <div class="reference">[30] Skula, L.: <b>Fermat's Last theorem and the Fermat quotients</b>. Comment. Math. Univ. St. Pauli 41 (1992), 35-54. <a href="http://www.ams.org/mathscinet-getitem?mr=1166223" target="_blank">MR 1166223</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0753.11016" target="_blank">Zbl 0753.11016</a></div> <div class="reference">[31] Skula, L.: <b>A note on some relations among special sums of reciprocals modulo $p$</b>. Math. Slovaca 58 (2008), 5-10. <a href="http://dx.doi.org/10.2478/s12175-007-0050-3" target="_blank">DOI 10.2478/s12175-007-0050-3</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2372821" target="_blank">MR 2372821</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1164.11001" target="_blank">Zbl 1164.11001</a></div> <div class="reference">[32] Slavutsky, I. S.: <b>Leudesdorf's theorem and Bernoulli numbers</b>. Arch. Math., Brno 35 (1999), 299-303. <a href="http://www.ams.org/mathscinet-getitem?mr=1744517" target="_blank">MR 1744517</a></div> <div class="reference">[33] Spivey, M. Z.: <b>Combinatorial sums and finite differences</b>. Discrete Math. 307 (2007), 3130-3146. <a href="http://dx.doi.org/10.1016/j.disc.2007.03.052" target="_blank">DOI 10.1016/j.disc.2007.03.052</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2370116" target="_blank">MR 2370116</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1129.05006" target="_blank">Zbl 1129.05006</a></div> <div class="reference">[34] Sun, Z. H.: <b>Congruences concerning Bernoulli numbers and Bernoulli polynomials</b>. Discrete Appl. Math. 105 (2000), 193-223. <a href="http://dx.doi.org/10.1016/S0166-218X(00)00184-0" target="_blank">DOI 10.1016/S0166-218X(00)00184-0</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1780472" target="_blank">MR 1780472</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0990.11008" target="_blank">Zbl 0990.11008</a></div> <div class="reference">[35] Sun, Z. H.: <b>Congruences involving Bernoulli and Euler numbers</b>. J. Number Theory 128 (2008), 280-312. <a href="http://dx.doi.org/10.1016/j.jnt.2007.03.003" target="_blank">DOI 10.1016/j.jnt.2007.03.003</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2380322" target="_blank">MR 2380322</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1154.11010" target="_blank">Zbl 1154.11010</a></div> <div class="reference">[36] Sun, Z. H.: <b>Five congruences for primes</b>. Fibonacci Q. 40 (2002), 345-351. <a href="http://www.ams.org/mathscinet-getitem?mr=1920576" target="_blank">MR 1920576</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1009.11004" target="_blank">Zbl 1009.11004</a></div> <div class="reference">[37] Sun, Z. H.: <b>The combinatorial sum $\sum\nolimits_{k=0 , k\equiv r \pmod{m}}^n \binom{n}{k}$ and its applications in number theory II</b>. J. Nanjing Univ., Math. Biq. 10 (1993), 105-118 Chinese. <a href="http://www.ams.org/mathscinet-getitem?mr=1248315" target="_blank">MR 1248315</a></div> <div class="reference">[38] Sun, Z. W.: <b>A congruence for primes</b>. Proc. Am. Math. Soc. 123 (1995), 1341-1346. <a href="http://dx.doi.org/10.1090/S0002-9939-1995-1242105-X" target="_blank">DOI 10.1090/S0002-9939-1995-1242105-X</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1242105" target="_blank">MR 1242105</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0833.11001" target="_blank">Zbl 0833.11001</a></div> <div class="reference">[39] Sun, Z. W.: <b>Binomial coefficients, Catalan numbers and Lucas quotients</b>. Sci. China, Math. 53 (2010), 2473-2488. <a href="http://dx.doi.org/10.1007/s11425-010-3151-3" target="_blank">DOI 10.1007/s11425-010-3151-3</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=2718841" target="_blank">MR 2718841</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1221.11054" target="_blank">Zbl 1221.11054</a></div> <div class="reference">[40] Sun, Z. W.: <b>On the sum $\sum\nolimits_{k\equiv r \pmod{m}} \binom{n}{k}$ and related congruences</b>. Isr. J. Math. 128 (2002), 135-156. <a href="http://dx.doi.org/10.1007/BF02785421" target="_blank">DOI 10.1007/BF02785421</a> | <a href="http://www.ams.org/mathscinet-getitem?mr=1910378" target="_blank">MR 1910378</a></div> <div class="reference">[41] Sylvester, J. J.: <b>Sur une propriété des nombres premiers qui se ratache au théoreme de Fermat</b>. 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).</div> <div class="reference">[42] Tauraso, R.: <b>Congruences involving alternating multiple harmonic sums</b>. Electron. J. Comb. 17 (2010), Research Paper R16, 11 pages. <a href="http://www.ams.org/mathscinet-getitem?mr=2587747" target="_blank">MR 2587747</a> | <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:1222.11006" target="_blank">Zbl 1222.11006</a></div> <div class="reference">[43] Wieferich, A.: <b>On Fermat's Last Theorem</b>. J. für Math. 136 (1909), 293-302 German.</div> <div class="reference">[44] Wolstenholme, J.: <b>On certain properties of prime numbers</b>. Quart. J. 5 (1862), 35-39.</div> </div> </div> </div> </div> <div id="ds-options"> <div id="search-box"> <h3>Search</h3> <div class="ds-option-set" id="ds-search-option"> <form method="get" id="ds-search-form" action="/search"> <fieldset> <input type="text" class="ds-text-field " name="query" /> <input value="Go" type="submit" name="submit" class="ds-button-field " onclick=" var radio = document.getElementById("ds-search-form-scope-container"); if (radio != undefined && radio.checked) { var form = document.getElementById("ds-search-form"); form.action= "/handle/" + radio.value + "/search" ; } " /> <label> <input checked="checked" value="" name="scope" type="radio" id="ds-search-form-scope-all" />Search</label> <br /> <label> <input name="scope" type="radio" id="ds-search-form-scope-container" value="10338.dmlcz/143603" />This Collection</label> </fieldset> </form> <a href="/advanced-search">Advanced Search</a> </div> </div> <div xmlns="http://www.w3.org/1999/xhtml" id="artifactbrowser_Navigation_list_browse" class="ds-option-set"> <ul class="ds-options-list"> <li> <h4 class="ds-sublist-head">Browse</h4> <ul class="ds-simple-list"> <li> <a href="/community-list">Collections</a> </li> <li> <a href="/browse-title">Titles</a> </li> <li> <a href="/browse-author">Authors</a> </li> <li> <a href="/MSCSubjects">MSC</a> </li> </ul> </li> </ul> </div> <div xmlns="http://di.tamu.edu/DRI/1.0/" class="ds-option-set" id="artifactbrowser_Navigation_list_account"> <h3 class="ds-option-set-head"> </h3> <a href="/about">About DML-CZ</a> </div> <div id="eudml-partner"> <div class="eudml-partner-head">Partner of</div> <a href="http://eudml.org/"> <img alt="EuDML logo" src="/manakin/themes/DML/eudml-logo-mensi.png" /> </a> </div> </div> <div id="ds-footer"> <div id="ds-footer-links">© 2010 <a href="http://www.math.cas.cz/">Institute of Mathematics CAS</a> </div> </div> </div> <script type="text/javascript"> var gaJsHost = (("https:" == document.location.protocol) ? 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