Products of Integers with Few Nonzero Digits

Hajime Kaneko; Thomas Stoll

doi:10.2478/udt-2022-0006

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Products of Integers with Few Nonzero Digits

Uniform distribution theory

Volume 17 (2022): Issue 1 (May 2022)

By: Hajime Kaneko and Thomas Stoll

Open Access

|May 2022

Abstract

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system

s(ab)= k, s(a)= ℓ, and s(b)= m

in odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of ℓ and m.

References

[1] BENNETT, M.—BUGEAUD, Y.: Perfect powers with three digits, Mathematika 60 (2014), no. 1, 66–84.
Search in Google Scholar Back to article
[2] BENNETT, M.—BUGEAUD, Y.—MIGNOTTE, M.: Perfect powers with few binary digits and related Diophantine problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 4, 941–953.
Search in Google Scholar Back to article
[3] BENNETT, M.—BUGEAUD, Y.—MIGNOTTE, M.: Perfect powers with few binary digits and related Diophantine problems II, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 3, 525–540.
Search in Google Scholar Back to article
[4] BEUKERS, F.: On the generalized Ramanujan-Nagell equation I,Acta Arith. 38 (1981), 389–410.10.4064/aa-38-4-389-410
Search in Google Scholar Back to article
[5] BRILLHART, J.—LEHMER, D. H.—SELFRIDGE, J. L.: New primality criteria and factorizations of 2^m± 1, Math. Comp. 29 (1975), 620–647.
Search in Google Scholar Back to article
[6] BUGEAUD, Y.—KANEKO, H.: On the digital representation of smooth numbers,Math. Proc. Cambridge Philos. Soc. 165 (2018), no. 3, 533–540.
Search in Google Scholar Back to article
[7] P. Corvaja, U. Zannier, Finiteness of odd perfect powers with four nonzero binary digits, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 2, 715–731.
Search in Google Scholar Back to article
[8] HARE, K. G.—LAISHRAM, S.—STOLL, T.: The sum of digits of n and n²,Int. J. Number Theory 7 (2011), no. 7, 1737–1752.
Search in Google Scholar Back to article
[9] HARE, K. G.—LAISHRAM, S.—STOLL, T.: Stolarsky’s conjecture and the sum of digits of polynomial values, Proc. Amer. Math. Soc. 139 (2011), no. 1, 39–49.
Search in Google Scholar Back to article
[10] LUCA, F.: The Diophantine equation x² = p^a± p^b +1, Acta Arith. 112 (2004), no. 1, 87–101.
Search in Google Scholar Back to article
[11] MADRITSCH, M.—STOLL, T.: On simultaneous digital expansions of polynomial values, Acta Math. Hungar. 143 (2014), no. 1, 192–200.
Search in Google Scholar Back to article
[12] MARTIN, B.—MAUDUIT, C.—RIVAT, J.: Propriétés locales des chiffres des nombres premiers, J. Inst. Math. Jussieu 18 (2019), no. 1, 189–224.
Search in Google Scholar Back to article
[13] MAUDUIT, C.—RIVAT, J.: Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. 2 171 (2010), no. (3), 1591–1646.
Search in Google Scholar Back to article
[14] MEI, S.-Y.: The sum of digits of polynomial values, Integers 15 (2015), Paper no. A 32, 12 pp.
Search in Google Scholar Back to article
[15] MELFI, G.: On simultaneous binary expansions of n and n²,J.Number Theory 111 (2005), 248–256.10.1016/j.jnt.2004.12.009
Search in Google Scholar Back to article
[16] SAUNDERS, J. C.: Sums of digits in q-ary expansions,Int.J.Number Theory 11 (2015), no. 2, 593–611.
Search in Google Scholar Back to article
[17] STEWART, C. L.: On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers,Proc. London Math. Soc. (3) 35 (1977), no. 3, 425–447.
Search in Google Scholar Back to article
[18] STOLARSKY, K. B.: The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), 1–5.10.1090/S0002-9939-1978-0495823-5
Search in Google Scholar Back to article
[19] SZALAY, L.: The equations 2ⁿ ± 2^m ± 2^l = z², Indag. Math. (N. S.) 13 (2002), no. 1, 131–142.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/udt-2022-0006 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 11 - 28

Submitted on: Jun 30, 2021

Accepted on: Nov 23, 2021

Published on: May 31, 2022

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Related subjects:

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2022 Hajime Kaneko, Thomas Stoll, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 17 (2022): Issue 1 (May 2022)