RESEARCH ARTICLE

Diophantine inequality involving binary forms

  • Quanwu MU
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  • School of Science, Xi’an Polytechnic University, Xi’an 710048, China

Received date: 26 Jul 2016

Accepted date: 24 Aug 2016

Published date: 27 Nov 2017

Copyright

2017 Higher Education Press and Springer-Verlag GmbH Germany

Abstract

Let d3 be an integer, and set r=2d1+1for3d4,r=17322d+1for5d6,r=d2+d+1for7d8, and r=d2+d+2for d9, respectively. Suppose that Φi(x,y)|x,y|(1ir) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2, . . . , λr are nonzero real numbers with λ12 irrational, and λ1Φ1 (x1, y1) + λ2Φ2 (x2, y2) + · · · + λrΦr (xr, yr) is indefinite. Then for any given real η and σ with 0<σ<22−d, it is proved that the inequality has infinitely many solutions in integers x1, x2, . . . , xr, y1, y2, . . . , yr. This result constitutes an improvement upon that of B. Q. Xue.

Cite this article

Quanwu MU . Diophantine inequality involving binary forms[J]. Frontiers of Mathematics in China, 2017 , 12(6) : 1457 -1468 . DOI: 10.1007/s11464-017-0602-y

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