Padovan or Perrin numbers that are concatenations of two distinct base b repdigits
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Kouèssi N. Adédji
Abstract
Let {Pn}n⩾0 be the Padovan sequence with initial conditions P0=0, P1=1, and P2=1 and the recurrence relation Pn+3=Pn+1 + Pn. Its companion sequence is known as the Perrin sequence {En}n⩾0 that satisfies the same above recurrence relation with the initial conditions E0=3, E1=0 and E2=2. In this paper, we determine all Padovan and Perrin numbers that are concatenations of two distinct base b repdigits with 2 ⩽ b ⩽ 9. As corollary, we prove that the largest Padovan and Perrin numbers which can be representable as a concatenations of two distinct base b repdigits are
Acknowledgement
We thank the referee for a careful reading of our manuscript and for useful comments.
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(Communicated by István Gaál)
References
[1] Adédji, K. N.—Filipin, A.—Togbé, A.: Padovan and Perrin numbers which are products of two repdigits in base b, Results Math. 78 (2021), Art. No. 193.10.1007/s00025-021-01502-6Search in Google Scholar
[2] Baker, A.—Davenport, H.: The equations 3x2−2=y2 and 8x2−7=z2, Quart. J. Math. Oxford Ser. 20(2) (1969), 129–137.10.1093/qmath/20.1.129Search in Google Scholar
[3] Batte, H.—Chalebgwa, T. P.—Ddamulira, M.: Perrin numbers that are concatenations of two distinct repdigits, Arab. J. Math. 11, 469–478.10.1007/s40065-022-00388-8Search in Google Scholar
[4] Bugeaud, Y.—Mignotte, M.—Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), 969–1018.10.4007/annals.2006.163.969Search in Google Scholar
[5] Ddamulira, M.: Repdigits as sums of three Padovan numbers, Bol. Soc. Mat. Mex. 26(2) 2020, 247–261.10.1007/s40590-019-00269-9Search in Google Scholar PubMed PubMed Central
[6] Ddamulira, M.: Padovan numbers that are concatenations of two distinct repdigits, Math. Slovaca 71(2) (2021), 275–284.10.1515/ms-2017-0467Search in Google Scholar
[7] Dujella, A.—Pethö, A.: A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxf. Ser. (2) 49 (1998), 291–306.10.1093/qjmath/49.195.291Search in Google Scholar
[8] García Lomelí, A. C.—Hernández Hernández, S.: Repdigits as sums of two Padovan numbers, J. Integer Seq. 22(2) (2019), Art. 19.2.3.Search in Google Scholar
[9] Guzmán, S.—Luca, F.: Linear combinations of factorials and s-units in a binary recurrence sequence, Ann. Math. Qué. 38 (2014), 169–188.Search in Google Scholar
[10] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers Ii, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180 (in Russian); Engl. translation in Izv. Math. 64 (2000), 1217–1269.Search in Google Scholar
[11] Murty, M. R.—Esmonde, J.: Problems in algebraic number theory, 2nd ed., Grad. Texts in Math. 190, Springer-Verlag, New York, 2005.10.1007/978-3-642-87939-5_1Search in Google Scholar
[12] Rihane, S. E.—Togbé, A.: Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math. 10 (2021), 469–480.10.1007/s40065-021-00317-1Search in Google Scholar
[13] Sloane et al.: The On-Line Encyclopedia of Integer Sequences, 2019. Available at http://oeis.org10.1515/9780691197944-009Search in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- RNDr. Stanislav Jakubec, DrSc. passed away
- Schur m-power convexity for general geometric Bonferroni mean of multiple parameters and comparison inequalities between several means
- Conditions forcing the existence of relative complements in lattices and posets
- Radically principal MV-algebras
- A topological duality for dcpos
- Padovan or Perrin numbers that are concatenations of two distinct base b repdigits
- Addendum to “A generalization of a result on the sum of element orders of a finite group”
- Remarks on w-distances and metric-preserving functions
- Solution of logarithmic coefficients conjectures for some classes of convex functions
- Multiple periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian and partially periodic potentials
- Asymptotic stability of nonlinear neutral delay integro-differential equations
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- (ε, A)-approximate numerical radius orthogonality and numerical radius derivative
- Wg-Drazin-star operator and its dual
- On the Lupaş q-transform of unbounded functions
- K-contact and (k, μ)-contact metric as a generalized η-Ricci soliton
- Compactness with ideals
- Number of cells containing a given number of particles in a generalized allocation scheme
- A new generalization of Nadarajah-Haghighi distribution with application to cancer and COVID-19 deaths data
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