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On integers with identical digits

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

Yann Bugeaud  and
Maurice Mignotte
Yann Bugeaud
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7. rue René Descartes, 67084 Strasbourg, France. E-mail: [email protected]
Maurice Mignotte
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg, France. E-mail: [email protected]
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Extract

A long-standing conjecture claims that the Diophantine equation

has finitely many solutions, and, maybe, only those given by

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 411 - 417
Copyright
Copyright © University College London 1999

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References

1. Bennett, M. and Weger, B. M. M. De. On the Diophantine equation . Math. Comp., 67 (1998), 413–438.CrossRefGoogle Scholar
2. Bugeaud, Y.. Linear forms in p-adic logarithms and the Diophantine equation . Math. Proc. Cambridge Philos,. Soc, 127 (1999), 373–381.CrossRefGoogle Scholar
3. Bugeaud, Y. and Laurent, M.. Minoration effective de la distance p-adique entre puissances de nombres algébriques. J. Number Th., 61 (1996), 311–342.CrossRefGoogle Scholar
4. Bugeaud, Y., Mignotte, M., Roy, Y. and Shoreyl, T. N.. The Diophantine equation has no solution with x square. Math. Proc. Cambridge Phitos. Soc. 127 (1999), 353–372.CrossRefGoogle Scholar
5. Cassels, J. W. S.. On the equation ax - by=1, II. Proc. Camb. Phil. Soc., 56 (1960), 97–103.CrossRefGoogle Scholar
6. Cresenzo, P.. A Diophantine equation which arises in the theory of finite groups. Adv. Math., 17 (1975), 25–29.CrossRefGoogle Scholar
7. Inkeri, K.. On the Diophantine equation . Acta Arith., 21 (1972), 299–311.CrossRefGoogle Scholar
8. Laurent, M., Mignotte, M. and Nesterenko, Y.. Formes linéaires en deux logarithmes et déterminants d'interpolation. J. Number Th., 55 (1995), 285–321.CrossRefGoogle Scholar
9. Le, M.. A note on the Diophantine equation . Acta Arith., 64 (1993). 19–28.CrossRefGoogle Scholar
10. Le, M.. A note on the Diophantine equation . Math. Proc. Cambridge Philos. Soc., 116 (1994), 385–389.Google Scholar
11. Le, M.. A Diophantine equation concerning finite groups. Pacific J. of Math., 169 (1995). 335–341.CrossRefGoogle Scholar
12. Le, M.. A note on perfect powers of the form 1, Acta Arith., 69 (1995). 91–98.CrossRefGoogle Scholar
13. Ljunggren, W.. Noen Setninger om ubestemte likninger av formen . Norsk. Mat. Tidsskr., 25 (1943), 17–20.Google Scholar
14. Nagell, T.. Note sur l'equation indéterminée . Norsk. Mat. Tidsskr., 2 (1920), 75–78.Google Scholar
15. Oblàth, R.. Une propriété des puissances parfaites. Mathesis, 65 (1956), 356–364.Google Scholar
16. Saradha, N. and Shorey, T. N.. The equation with x square. Math. Proc. Cambridge Philos. Soc., 125 (1999), 1–19.CrossRefGoogle Scholar
17. Shorey, T. N.. Exponential Diophantine equations involving product of consecutive integers and related equations. Number Theory, Trends Math. (Birkhäuser, Basel, 2000), 463–495.Google Scholar
18. Shorey, T. N. and Tijdeman, R.. New applications of Diophantine approximations to Diophantine equations. Math. Scand., 39 (1976), 5–18.CrossRefGoogle Scholar
19. Shorey, T. N. and Tijdeman, R.. Exponential Diophantine equations, Cambridge Tracts in Mathematics 87 (Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
20. Sun, Q.. An unsolved Diophantine equation in finite groups. J. Math. Res. Exposition, 6 (1986), 20 (Chinese).Google Scholar
21. Yu, L. and Le, M.. On the Diophantine equation . Acta Arith., 83 (1995), 363–366.CrossRefGoogle Scholar
22. Yuan, P.-Z.. Comment: A note on perfect powers of the form . Acta Arith., 69. (1995), 91–98, by M. Le and On the Diophantine equation . Acta Arith., 83 (1995), 363-366, by L. Yu and M. Le. Acta Arith., 83 (1998), 199.Google Scholar