The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations

References

• [Bennett 01] M. Bennett. “Rational Approximation to Algebraic Numbers of Small Height: The Diophantine Equation axn − byn| = 1|.” J. Reine Angew. Math. 535 (2001), 1–49.
   Web of Science ®Google Scholar
• [Bennett and Levin 15] M. Bennett and A. Levin. “The Nagell-Ljunggren Equation Via Runge’s Method.” Monatsh. Math. 177 (2015), 15–31.
   Web of Science ®Google Scholar
• [Berstel 95] J. Berstel. “Axel Thue’s Papers on Repetitions in Words: A Translation.” Number 20 in Publications du Laboratoire de Combinatoire et d’Informatique Mathématique. Université du Québec à Montréal, February 1995.
   Google Scholar
• [Boute 00] R. T. Boute. “Zeroless Positional Number Representation and String Ordering.” Amer. Math. Monthly 107 (2000), 437–444.
   Web of Science ®Google Scholar
• [Broughan 12] K. A. Broughan. “An Explicit Bound for Aliquot Cycles of Repdigits.” INTEGERS 12 (2012), #A15 (electronic).
   Google Scholar
• [Browkin 00] J. Browkin. “The abc-Conjecture.” Number Theory, Trends Math., pp. 75–105. Boston: Birkhäuser, 2000.
   Google Scholar
• [Browkin 08] J. Browkin. “A Weak Effective abc-Conjecture.” Funct. Approx. Comment. Math. 39 (2008), 103–111.
   Google Scholar
• [Bugeaud 02] Y. Bugeaud. “Linear Forms in Two m-adic Logarithms and Applications to Diophantine Problems.” Compositio Math. 132 (2002), 137–158.
   Web of Science ®Google Scholar
• [Bugeaud et al. 02] Y. Bugeaud, G. Hanrot and M. Mignotte. “Sur l’équation diophantienne (xn − 1)/(x − 1) = yq. III.” Proc. Lond. Math. Soc. 84 (2002), 59–78.
   Web of Science ®Google Scholar
• [Bugeaud and Mignotte 99a] Y. Bugeaud and M. Mignotte. “On Integers with Identical Digits.” Mathematika 46 (1999a), 411–417.
   Web of Science ®Google Scholar
• [Bugeaud and Mignotte 99b] Y. Bugeaud and M. Mignotte. “Sur l’équation diophantienne (xn − 1)/(x − 1) = yq. II.” C. R. Acad. Sci. Paris 328 (1999b), 741–744.
   Google Scholar
• [Bugeaud and Mignotte 02] Y. Bugeaud and M. Mignotte. “L’équation de Nagell-Ljunggren xn-1x-1=yq.” Enseign. Math. 48 (2002), 147–168.
   Google Scholar
• [Bugeaud et al. 00] Y. Bugeaud, M. Mignotte, and Y. Roy. “On the Diophantine Equation (xn − 1)/(x − 1) = yq.” Pacific J. Math. 193 (2000), 257–268.
   Web of Science ®Google Scholar
• [Bugeaud et al. 99] Y. Bugeaud, M. Mignotte, Y. Roy, and T. N. Shorey. “The Equation (xn − 1)/(x − 1) = yq has no Solution with x Square.” Math. Proc. Cambridge Phil. Soc. 127 (1999), 353–372.
   Web of Science ®Google Scholar
• [Bugeaud and Mihăilescu 07] Y. Bugeaud and P. Mihăilescu. “On the Nagell-Ljunggren Equation xn-1x-1=yq.” Math. Scand. 101 (2007), 177–183.
   Web of Science ®Google Scholar
• [Cilleruelo et al. 09] J. Cilleruelo, F. Luca, and I. E. Shparlinski. “Power Values of Palindromes.” J. Combin. Number Theory 1 (2009), 101–107.
   Google Scholar
• [Davis and Weyuker 83] M. D. Davis and E. J. Weyuker. Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science. New York: Academic Press, 1983.
   Google Scholar
• [Forslund 95] R. R. Forslund. “A Logical Alternative to the Existing Positional Number System.” Southwest J. Pure Appl. Math. 1 (1995), 27–29.
   Google Scholar
• [Foster 47] J. E. Foster. “A number System Without a Zero Symbol.” Math. Mag. 21:1 (1947), 39–41.
   Google Scholar
• [Granville 07] A. Granville. “Rational and Integral Points on Quadratic Twists of a Given Hyperelliptic Curve.” Int. Math. Res. Not. IMRN, No. 8, (2007), Art. ID 027.
   Google Scholar
• [Granville and Tucker 02] A. Granville and T. J. Tucker. “It’s as Easy as abc.” Notices Amer. Math. Soc. 49 (2002), 1224–1231.
   Google Scholar
• [Hernández and Luca 06] S. Hernández and F. Luca. “Palindromic Powers.” Rev. Colombiana Mat. 40 (2006), 81–86.
   Google Scholar
• [Hirata-Kohno and Shorey 97] N. Hirata-Kohno and T. N. Shorey. “On the Equation (xm − 1)/(x − 1) = yq with x Power.” , London Math. Soc. Lecture Note Series, vol. 247, edited by Y. Motohashi, pp. 343–351. Cambridge: Cambridge University Press, 1997.
   Google Scholar
• [Jacobson and Williams 09] M. J. Jacobson, Jr., and H. C. Williams. Solving the Pell Equation. CMS Books in Mathematics, New York: Springer, 2009.
   Google Scholar
• [Kihel 09] O. Kihel. “A Note on the Nagell-Ljunggren Diophantine Equation xn-1x-1=yq.” JP J. Algebra Number Theory Appl. 13 (2009), 131–135.
   Google Scholar
• [Knuth 69] D. E. Knuth. The Art of Computer Programming. Volume 2: Seminumerical Algorithms. First edition. Reading, Mass: Addison-Wesley, 1969.
   Google Scholar
• [Korec 91] I. Korec. “Palindromic Squares for Various Number System Bases.” Math. Slovaca 41 (1991), 261–276.
   Google Scholar
• [Laishram and Shorey 12] S. Laishram and T. N. Shorey. “Baker’s Explicit abc-Conjecture and Applications.” Acta Arith. 155 (2012), 419–429.
   Web of Science ®Google Scholar
• [Le 94] M. H. Le. “A Note on the Diophantine Equation xm-1x-1=yn.” Math. Proc. Cambridge Phil. Soc. 116 (1994), 385–389.
   Web of Science ®Google Scholar
• [Lekkerkerker 52] C. G. Lekkerkerker. “Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci.” Simon Stevin 29 (1952), 190–195.
   Google Scholar
• [Li and Li 14] J. Li and X. Li. “On the Nagell-Ljunggren Equation and Edgar’s Conjecture.” Int. J. Appl. Math. Stat. 52 (2014), 80–83.
   Google Scholar
• [Ljunggren 43a] W. Ljunggren. “Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante.” Acta Math. 75 (1943a), 1–21.
   Google Scholar
• [Ljunggren 43b] W. Ljunggren. “Noen setninger om ubestemte likninger av formen (xn − 1)/(x − 1) = yq.” Norsk. Mat. Tidsskr. 25 (1943b), 17–20.
   Google Scholar
• [Masser 85] D. W. Masser. “Problem.” p. 25. Book of Abstracts of Conference. London: Department of Mathematics, Imperial College, 1985.
   Google Scholar
• [Mihăilescu 07] P. Mihăilescu. “New Bounds and Conditions for the Equation of Nagell-Ljunggren.” J. Number Theory 124 (2007), 380–395.
   Web of Science ®Google Scholar
• [Mihăilescu 08] P. Mihăilescu. “Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren.” Dev. Math., vol. 16, pp. 245–273. New York: Springer-Verlag, 2008.
   Google Scholar
• [Nagell 20] T. Nagell. “Note sur l’équation indéterminée (xn − 1)/(x − 1) = yq.” Norsk. Mat. Tidsskr. 2 (1920), 75–78.
   Google Scholar
• [Nagell 21] T. Nagell. “Des équations indéterminées x2 + x + 1 = yn et x2 + x + 1 = 3yn.” Norsk Mat. Forenings Skrifter (1921).
   Google Scholar
• [Nitaj 96] A. Nitaj. “La conjecture abc.” Enseign. Math. 42 (1996), 3–24.
   Google Scholar
• [Obláth 56] R. Obláth. “Une propriété des puissances parfaites.” Mathesis 65 (1956), 356–364.
   Google Scholar
• [Sloane et al. 17] N. J. A. Sloane, et al. The On-Line Encyclopedia of Integer Sequences. (2017). Available online (https://oeis.org).
   Google Scholar
• [Oesterlé 88] J. Oesterlé. “Nouvelles approches du “théorème” de Fermat.” Astérisque 161--(1988), 165–186. Séminaire Bourbaki, exp. 694.
   Google Scholar
• [Robert et al. 14] O. Robert, C. Stewart, and G. Tenenbaum. “A Refinement of the abc Conjecture.” Bull. Lond. Math. Soc. 46 (2014), 1156–1166.
   Web of Science ®Google Scholar
• [Salomaa 73] A. Salomaa. Formal Languages. Cambridge, MA: Academic Press, 1973.
   Google Scholar
• [Shorey 86] T. N. Shorey. “On the Equation zq = (xn − 1)/(x − 1).” Indag. Math. 48 (1986), 345–351.
   Google Scholar
• [Shorey 00] T. N. Shorey. “Some Conjectures in the Theory of Exponential Diophantine Equations.” Publ. Math. (Debrecen) 56 (2000), 631–641.
   Web of Science ®Google Scholar
• [Smullyan 61] R. M. Smullyan. Theory of Formal Systems, Annals of Mathematical Studies. vol. 47. Princeton Princeton University Press, 1961.
   Google Scholar
• [Stewart and Tijdeman 86] C. L. Stewart and R. Tijdeman. “On the Oesterlé-Masser Conjecture.” Monatsh. Math. 102 (1986), 251–257.
   Web of Science ®Google Scholar
• [Thue 06] A. Thue. “Über unendliche Zeichenreihen.” , 1–22. Reprinted in Selected Mathematical Papers of Axel Thue, edited by T. Nagell (Oslo: Universitetsforlaget, 1977), 139–158.
   Google Scholar
• [Thue 12] A. Thue. “Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen.” Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912), 1–67. Reprinted in Selected Mathematical Papers of Axel Thue, edited by T. Nagell (Oslo: Universitetsforlaget, 1977), 413–478.
   Google Scholar
• [Yates 78] S. Yates. “The Mystique of Repunits.” Math. Mag. 51 (1978), 22–28.
   Google Scholar
• [Zeckendorf 72] E. Zeckendorf. “Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres Lucas.” Bull. Soc. Roy. Liége 41 (1972), 179–182.
   Google Scholar