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On the Diophantine equation $ 1+2\sp a=3\sp b5\sp c+2\sp d3\sp e5\sp f$


Author: Leo J. Alex
Journal: Math. Comp. 44 (1985), 267-278
MSC: Primary 11D61
MathSciNet review: 771050
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Abstract: In this paper the Diophantine equation $ 1 + {2^a} = {3^b}{5^c} + {2^d}{3^e}{5^f}$, where a, b, c, d, e and f are nonnegative integers, is solved. The related equations $ 1 + {3^a} = {2^b}{5^c} + {2^d}{3^e}{5^f}$ and $ 1 + {5^a} = {2^b}{3^c} + {2^d}{3^e}{5^f}$ are also solved. This work is related to and extends recent work of L. L. Foster, J. L. Brenner, and the author.


References [Enhancements On Off] (What's this?)

  • [1] Leo J. Alex, Diophantine equations related to finite groups, Comm. Algebra 4 (1976), no. 1, 77–100. MR 0424675
  • [2] L. J. Alex, "The Diophantine equation $ {3^a} + {5^b} = {7^c} + {11^d}$," Notices Amer. Math. Soc., v. 26, 1979, p. A-454, Abstract #768-10-13.
  • [3] Leo J. Alex, Simple groups and a Diophantine equation, Pacific J. Math. 104 (1983), no. 2, 257–262. MR 684288
  • [4] Leo J. Alex and Lorraine L. Foster, On Diophantine equations of the form 1+2^{𝑎}=𝑝^{𝑏}𝑞^{𝑐}+2^{𝑑}𝑝^{𝑒}𝑞^{𝑓}, Rocky Mountain J. Math. 13 (1983), no. 2, 321–331. MR 702828, 10.1216/RMJ-1983-13-2-321
  • [5] J. L. Brenner and Lorraine L. Foster, Exponential Diophantine equations, Pacific J. Math. 101 (1982), no. 2, 263–301. MR 675401
  • [6] Eugène Dubois and Georges Rhin, Sur la majoration de formes linéaires à coefficients algébriques réels et 𝑝-adiques. Démonstration d’une conjecture de K. Mahler, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 21, Ai, A1211–A1214. MR 0429783
  • [7] Hans Peter Schlickewei, Über die diophantische Gleichung 𝑥₁+𝑥₂\cdots+𝑥_{𝑛}=0, Acta Arith. 33 (1977), no. 2, 183–185. MR 0439747
  • [8] H. G. Senge and E. G. Straus, 𝑃𝑉-numbers and sets of multiplicity, Period. Math. Hungar. 3 (1973), 93–100. Collection of articles dedicated to the memory of Alfréd Rényi, II. MR 0340185
  • [9] R. Tijdeman, "Diophantine equations (Approximation methods)." (To appear.)
  • [10] Chen Te Yen, Some exponential Diophantine equations, Bull. Inst. Math. Acad. Sinica 13 (1985), no. 1, 49–92. MR 794914

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0771050-0
Keywords: Exponential Diophantine equation
Article copyright: © Copyright 1985 American Mathematical Society