Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on the exponential Diophantine equation $ x\sp 2-2\sp m=y\sp n$

Authors: Yongdong Guo and Mao Hua Le
Journal: Proc. Amer. Math. Soc. 123 (1995), 3627-3629
MSC: Primary 11D61
MathSciNet review: 1291786
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we prove that the equation $ {x^2} - {2^m} = {y^n},x,y,m,n \in \mathbb{N},\gcd (x,y) = 1,y > 1,n > 2$, has only finitely many solutions (x, y, m, n). Moreover, all solutions of the equation satisfy $ 2\nmid mn,n < 2 \cdot {10^9}$ and $ \max (x,y,m) < C$, where C is an effectively computable absolute constant.

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

  • [1] P.-P. Dong, Minoration de combinaisons linéaires de deux logarithmes p-adiques, Ann. Fac. Sci. Toulouse 12 (1991), 195-250. MR 1189441 (93i:11080)
  • [2] S. V. Kotov, Über die maximale norm der idealteiler des polynoms $ \alpha {x^m} + \beta {y^n}$ mit algebraischen Koeffizienten, Acta Arith. 31 (1976), 219-230. MR 0427226 (55:261)
  • [3] S. Rabinowitz, The solution of $ {y^2} \pm {2^n} = {x^3}$, Proc. Amer. Math. Soc. 62 (1976), 1-6. MR 0424678 (54:12637)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11D61

Retrieve articles in all journals with MSC: 11D61

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society