Mersenne numbers as a difference of two Lucas numbers

Murat Alan

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 3, page 269-276
  • ISSN: 0010-2628

Abstract

top
Let ( L n ) n ≥ 0 be the Lucas sequence. We show that the Diophantine equation L n - L m = M k has only the nonnegative integer solutions ( n , m , k ) = ( 2 , 0 , 1 ) , ( 3 , 1 , 2 ) , ( 3 , 2 , 1 ) , ( 4 , 3 , 2 ) , ( 5 , 3 , 3 ) , ( 6 , 2 , 4 ) , ( 6 , 5 , 3 ) where M k = 2 k - 1 is the k th Mersenne number and n > m .

How to cite

top

Alan, Murat. "Mersenne numbers as a difference of two Lucas numbers." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 269-276. <http://eudml.org/doc/299043>.

@article{Alan2022,
abstract = {Let $(L_n)_\{n\ge 0\}$ be the Lucas sequence. We show that the Diophantine equation $ L_n-L_m=M_k$ has only the nonnegative integer solutions $(n,m,k)= (2,0,1)$, $(3, 1, 2)$, $(3, 2, 1)$, $(4, 3, 2)$, $(5, 3, 3)$, $(6, 2, 4)$, $(6, 5, 3)$ where $ M_k=2^k-1 $ is the $k$th Mersenne number and $ n > m$.},
author = {Alan, Murat},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lucas number; Mersenne number; Diophantine equation; linear forms in logarithm},
language = {eng},
number = {3},
pages = {269-276},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Mersenne numbers as a difference of two Lucas numbers},
url = {http://eudml.org/doc/299043},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Alan, Murat
TI - Mersenne numbers as a difference of two Lucas numbers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 269
EP - 276
AB - Let $(L_n)_{n\ge 0}$ be the Lucas sequence. We show that the Diophantine equation $ L_n-L_m=M_k$ has only the nonnegative integer solutions $(n,m,k)= (2,0,1)$, $(3, 1, 2)$, $(3, 2, 1)$, $(4, 3, 2)$, $(5, 3, 3)$, $(6, 2, 4)$, $(6, 5, 3)$ where $ M_k=2^k-1 $ is the $k$th Mersenne number and $ n > m$.
LA - eng
KW - Lucas number; Mersenne number; Diophantine equation; linear forms in logarithm
UR - http://eudml.org/doc/299043
ER -

References

top
  1. Baker A., Davenport H., The equations 3 x 2 - 2 = y 2 and 8 x 2 - 7 = z 2 , Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR0248079
  2. Bravo J. J., Gómez C. A., 10.3336/gm.51.2.02, Glas. Matemat. Ser. III 51(71), (2016), no. 2, 307–319. MR3580200DOI10.3336/gm.51.2.02
  3. Bravo J. J., Luca F., 10.5486/PMD.2013.5390, Publ. Math. Debrecen 82 (2013), no. 3–4, 623–639. MR3066434DOI10.5486/PMD.2013.5390
  4. Bravo J. J., Luca F., Powers of two as sums of two Lucas numbers, J. Integer Seq. 17 (2014), no. 8, Article 14.8.3, 12 pages. MR3248227
  5. Demirtürk Bitim B., 10.1007/s10998-019-00287-0, Period. Math. Hungar. 79 (2019), no. 2, 210–217. MR4022203DOI10.1007/s10998-019-00287-0
  6. Dujella A., Pethö A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 195, 291–306. MR1645552
  7. Erduvan F., Keskin R., 10.3906/mat-1810-83, Turkish J. Math. 43 (2019), no. 3, 1115–1123. MR3962520DOI10.3906/mat-1810-83
  8. Hardy G. H., Wright E. M., An Introduction to the Theory of Numbers, The Clarendon Press, Oxford University Press, New York, 1979. MR0568909
  9. Kebli S., Kihel O., Larone J., Luca F., 10.1016/j.jnt.2020.08.004, J. Number Theory 220 (2021), 107–127. MR4177538DOI10.1016/j.jnt.2020.08.004
  10. Koshy T., Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001. MR1855020
  11. 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), no. 6, 125–180 (Russian); translation in Izv. Math. 64 (2000), no. 6, 1217–1269. MR1817252
  12. Normenyo B. V., Luca F., Togbé A., Repdigits as sums of four Fibonacci or Lucas numbers, J. Integer Seq. 21 (2018), no. 7, Art. 18.7.7, 30 pages. MR3858063
  13. Şiar Z., Keskin R., 10.4064/cm7485-12-2018, Colloq. Math. 159 (2020), no. 1, 119–126. MR4036721DOI10.4064/cm7485-12-2018
  14. Trojovský P., 10.18514/MMN.2018.1750, Miskolc Math. Notes 19 (2018), no. 1, 641–648. MR3895605DOI10.18514/MMN.2018.1750

NotesEmbed ?

top

You must be logged in to post comments.