Two exponential diophantine equations

Dominik J. Leitner[1]

  • [1] Mathematisches Institut Universität Basel Rheinsprung 21 4051 Basel, Switzerland

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 2, page 479-487
  • ISSN: 1246-7405

Abstract

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The equation 3 a + 5 b - 7 c = 1 , to be solved in non-negative rational integers a , b , c , has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation y 2 = 3 a + 2 b + 1 , to be solved in non-negative rational integers a , b and a rational integer y , has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact only six.

How to cite

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Leitner, Dominik J.. "Two exponential diophantine equations." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 479-487. <http://eudml.org/doc/219785>.

@article{Leitner2011,
abstract = {The equation $3^a+5^b-7^c=1$, to be solved in non-negative rational integers $a,b,c$, has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation $y^2=3^a+2^b+1$, to be solved in non-negative rational integers $a,b$ and a rational integer $y$, has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact only six.},
affiliation = {Mathematisches Institut Universität Basel Rheinsprung 21 4051 Basel, Switzerland},
author = {Leitner, Dominik J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {exponential Diophantine equations},
language = {eng},
month = {6},
number = {2},
pages = {479-487},
publisher = {Société Arithmétique de Bordeaux},
title = {Two exponential diophantine equations},
url = {http://eudml.org/doc/219785},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Leitner, Dominik J.
TI - Two exponential diophantine equations
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 479
EP - 487
AB - The equation $3^a+5^b-7^c=1$, to be solved in non-negative rational integers $a,b,c$, has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation $y^2=3^a+2^b+1$, to be solved in non-negative rational integers $a,b$ and a rational integer $y$, has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact only six.
LA - eng
KW - exponential Diophantine equations
UR - http://eudml.org/doc/219785
ER -

References

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  9. S. Lang, Fundamentals of Diophantine Geometry. Springer, 1983. Zbl0528.14013MR715605
  10. D. W. Masser, Mixing and linear equations over groups in positive characteristic. Israel Journal of Mathematics 142 (2004), 189–204. Zbl1055.37009MR2085715
  11. P. Vojta, A more general a b c conjecture. International Mathematics Research Notices 21 (1998), 1103–1116. Zbl0923.11059MR1663215
  12. U. Zannier, Some applications of diophantine approximation to diophantine equations. Forum Editrice Universitaria Udinese S.r.l. (2003). 
  13. U. Zannier, Polynomial squares of the form a X m + b ( 1 - X ) n + c . Rendiconti del Seminario Matematico della Università di Padova 112 (2004), 1–9. Zbl1167.11304MR2109949
  14. U. Zannier, Diophantine equations with linear recurrences. An overview of some recent progress. Journal de Théorie des Nombres de Bordeaux 17 (2005), 432–435. Zbl1162.11330MR2152233

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