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On the diophantine equation

✍ Scribed by Pingzhi Yuan; Yongzhong Hu

Publisher: Elsevier Science
Year: 2005
Tongue: English
Weight: 192 KB
Volume: 111
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2004.11.005

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✦ Synopsis

Let D > 2 be a positive integer, and let p be an odd prime not dividing D. In this paper, using the deep result of Bilu, Hanrot and Voutier (i.e., the existence of primitive prime factors of Lucas and Lehmer sequences), by computing Jacobi's symbols and using elementary arguments, we prove that: if (D, p) = (4, 5), (2, 5), then the diophantine equation x 2 + D m = p n has at most two positive integer solutions (x, m, n). Moreover, both x 2 + 4 m = 5 n and x 2 + 2 m = 5 n have exactly three positive integer solutions (x, m, n).

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