An elementary proof of a congruence by Skula and Granville

Romeo Meštrović

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 2, page 113-120
  • ISSN: 0044-8753

Abstract

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Let p ≥ 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 ≡ - ∑ k = 1 p - 1 2 k k 2 ( mod p ) . In this note we establish the above congruence by entirely elementary number theory arguments.

How to cite

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Meštrović, Romeo. "An elementary proof of a congruence by Skula and Granville." Archivum Mathematicum 048.2 (2012): 113-120. <http://eudml.org/doc/247159>.

@article{Meštrović2012,
abstract = {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.},
author = {Meštrović, Romeo},
journal = {Archivum Mathematicum},
keywords = {congruence; Fermat quotient; harmonic numbers; congruence; Fermat quotient; harmonic numbers},
language = {eng},
number = {2},
pages = {113-120},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {An elementary proof of a congruence by Skula and Granville},
url = {http://eudml.org/doc/247159},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Meštrović, Romeo
TI - An elementary proof of a congruence by Skula and Granville
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 2
SP - 113
EP - 120
AB - 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.
LA - eng
KW - congruence; Fermat quotient; harmonic numbers; congruence; Fermat quotient; harmonic numbers
UR - http://eudml.org/doc/247159
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. Cao, H. Q., Pan, H., 10.1016/j.jnt.2006.02.004, J. Number Theory 121 (2006), 224–233. (2006) MR2274904DOI10.1016/j.jnt.2006.02.004
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  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. (1900) 
  5. Granville, A., Some conjectures related to Fermat’s Last Theorem, Number Theory (Banff, AB, 1988), de Gruyter, Berlin (1990), 177–192. (1990) Zbl0702.11015MR1106660
  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. (1997) Zbl0903.11005MR1483922
  7. Granville, A., The square of the Fermat quotient, Integers 4 (2004), # A22. (2004) Zbl1083.11005MR2116007
  8. Hardy, G. H., Wright, E. M., An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1960. (1960) Zbl0086.25803
  9. Lehmer, E., On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. (1938), 350–360. (1938) Zbl0019.00505MR1503412
  10. Ribenboim, P., 13 lectures on Fermat’s last theorem, Springer–Verlag, New York, Heidelberg, Berlin, 1979. (1979) Zbl0456.10006MR0551363
  11. Slavutsky, I. Sh., Leudesdorf’s theorem and Bernoulli numbers, Arch. Math. 35 (1999), 299–303. (1999) 
  12. Sun, Z. H., 10.1016/S0166-218X(00)00184-0, Discrete Appl. Math. 105 (1–3) (2000), 193–223. (2000) Zbl0990.11008MR1780472DOI10.1016/S0166-218X(00)00184-0
  13. Sun, Z. W., 10.1090/S0002-9939-2011-10925-0, Proc. Amer. Math. Soc. 140 (2012), 415–428. (2012) MR2846311DOI10.1090/S0002-9939-2011-10925-0
  14. Tauraso, R., Congruences involving alternating multiple harmonic sums, Electron. J. Comb. 17 (2010), # R16. (2010) Zbl1222.11006MR2587747
  15. Wolstenholme, J., On certain properties of prime numbers, Quart. J. Pure Appl. Math. 5 (1862), 35–39. (1862) 

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