On the finiteness of some p-divisible sets

Year 2024, Volume: 73 Issue: 4, 1011 - 1039

Çağatay Altuntaş

https://doi.org/10.31801/cfsuasmas.1441894

Abstract

For any positive integer $n$, let $H_n$ denote the $n^{th}$ harmonic number. Given a prime number $p$, it is not known whether the set of integers $J(p) = \{n \in \mathbb{N} : p \mid H_n \} $ is finite. In this paper, we first investigate a variant of this set, namely, we work on the divisibility properties of the differences of harmonic numbers. For any prime $p$ and a positive integer $w$, we define the set $D(p,w)$ as $\{n \in \mathbb{N} : p \mid H_n - H_w \}$ and work on the structure of this set. We present some finiteness results on $D(p,w)$ and obtain upper bounds for the number of elements in the set. Next, we consider the differences of generalized harmonic numbers and present an upper bound for the corresponding counting function. Moreover, under some plausible conditions, we prove that the difference set of generalized harmonic numbers is finite. Finally, we point out some directions to pursue.

Keywords

Harmonic numbers, generalized harmonic numbers, p-adic valuation

References

• Alkan, E., Approximation by special values of harmonic zeta function and logsine integrals, Commun. Number Theory Phys., 7(3) (2013), 515-550. https://doi.org/10.4310/cntp.2013.v7.n3.a5
• Alkan, E., Special values of the Riemann zeta function capture all real numbers, Proc. Amer. Math. Soc., 143(9) (2015), 3743-3752. https://doi.org/10.1090/s0002-9939-2015-12649-4
• Alkan, E., Göral, H., Sertbaş, D. C., Hyperharmonic numbers can rarely be integers, Integers, 18 (2018), A43. https://doi.org/10.5281/zenodo.10677684
• Altuntaş, Ç., Göral, H., Dedekind harmonic numbers, Proc. Indian Acad. Sci. Math. Sci., 131(2) (2021), 46-63. https://doi.org/10.1007/s12044-021-00643-6
• Altuntaş Ç., On the p-adic valuation of generalized harmonic numbers, Bull. Korean Math. Soc., 60(4) (2023), 933-955. https://doi.org/10.4134/BKMS.b220399
• Altuntaş, Ç., Göral H., Sertbaş, D. C., The difference of hyperharmonic numbers via geometric and analytic methods, J. Korean Math. Soc., 59(6) (2022), 1103-1137. https://doi.org/10.4134/JKMS.j210630
• Amrane, R. A., Belbachir, H., Nonintegerness of class of hyperharmonic numbers, Ann. Math. Inform., 37 (2010), 7-10.
• Amrane, R. A., Belbachir, H., Are the hyperharmonics integral? A partial answer via the small intervals containing primes, C. R. Math. Acad. Sci. Paris, 60(3) (2011), 115-117. https://doi.org/10.1016/j.crma.2010.12.015
• Babbage, C., Demonstration of a theorem relating to prime numbers, Edinburgh Philosophical J., 1 (1819), 46-49.
• Boyd, D. W., A p-adic study of the partial sums of the harmonic series, Experiment. Math., 18 (1994). https://doi.org/10.1080/10586458.1994.10504298
• Carlitz, L., A note on Wolstenholme's theorem, Amer. Math. Monthly, 61 (1954), 174-176. https://doi.org/10.2307/2307217
• Chen, Y. G., Wu, B. L., On generalized harmonic numbers, Combinatorial and additive number theory IV, Springer Proc. Math. Stat., 347 (2021), 107-129. https://doi.org/10.1007/978-3-030-67996-5_6
• Conway, J. H., Guy, R. K., The Book of Numbers, Springer-Verlag, New York, NY, USA, 1 edition 1996. https://doi.org/10.1007/978-1-4612-4072-3
• Çelik, S¸. C¸ ., Göral, H., Approximation by special values of Dirichlet series, Proc. Amer. Math. Soc., 148 (2020), 83-93. https://doi.org/10.1090/proc/14715
• Erdös, P., Niven, I., Some properties of partial sums of the harmonic series, Bull. Amer. Math. Soc., 52(4) (1946), 248-251. https://doi.org/10.1090/s0002-9904-1946-08550-x
• Eswarathasan, A., Levine, E., p-integral harmonic sums, Discrete Math., 91(3) (1991), 249-257. https://doi.org/10.1016/0012-365x(90)90234-9
• Gessel, I. M., Wolstenholme revisited, Amer. Math. Monthly, 105(7) (1998), 657-658. https://doi.org/10.2307/2589252
• Göral, H., Sertbaş, D. C., Almost all hyperharmonic numbers are not integers, J. Number Theory, 171 (2017), 495-526. https://doi.org/10.1016/j.jnt.2016.07.023https://doi.org/10.1016/j.jnt.2016.07.023
• Göral, H., Sertbaş, D. C., Divisibility properties of hyperharmonic numbers, Acta Math. Hungar., 154 (2018), 147-186. https://doi.org/10.1007/s10474-017-0766-7
• Göral, H., Sertbaş, D. C., A congruence for some generalized harmonic type sums, Int. J. Number Theory, 14(4) (2018), 1033-1046. https://doi.org/10.1142/s1793042118500628
• Göral, H., Sertbaş, D. C., Euler sums and non-integerness of harmonic type sums, Hacet. J. Math. Stat., 49(2) (2020), 586-598. https://doi.org/10.15672/hujms.544489
• Göral, H., Sertbaş, D. C., Applications of class numbers and Bernoulli numbers to harmonic type sums, Bull. Korean Math. Soc., 58(6) (2021), 1463-1481. https://doi.org/10.4134/BKMS.b201045
• Kürschak, J., On the harmonic series, Matematikai es Fizikai Lapok., 27 (1918), 299-300, (In Hungarian).
• Lengyel, T., On divisibility properties of some differences of the central binomial coefficients and catalan numbers, Integers, 13 (2013), A10. https://doi.org/10.1515/9783110298161.129
• Mezö, I., About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 50 (2007), 13-20. http://dx.doi.org/10.48550/arXiv.0811.0043
• SageMath, the Sage Mathematics Software System (Version 8.3), The Sage Developers, (2018). https://www.sagemath.org
• Sanna, C., On the p-adic valuation of harmonic numbers, J. Number Theory, 166 (2016), 41-46. https://doi.org/10.1016/j.jnt.2016.02.020
• Sertbaş, D. C., Hyperharmonic integers exist, C. R. Math. Acad. Sci. Paris, 358(11-12) (2020), 1179-1185. https://doi.org/10.5802/crmath.137
• Wolstenholme, J., On certain properties of prime numbers, Quart. J. Pure Appl. Math., 5 (1862), 35-39.
• Wu, B. L., Chen, Y. G., On certain properties of harmonic numbers, J. Number Theory, 175 (2017), 66-86. https://doi.org/10.1016/j.jnt.2016.11.027