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On the Diophantine Equation x2+q2k+1=yn

✍ Scribed by S.A. Arif; Fadwa S. Abu Muriefah

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
89 KB
Volume
95
Category
Article
ISSN
0022-314X

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✦ Synopsis

In this paper it has been proved that if q is an odd prime, qc7 ðmod 8Þ; n is an odd integer 55, n is not a multiple of 3 and ðh; nÞ ¼ 1, where h is the class number of the filed Qð ffiffiffiffiffiffi ffi Àq p Þ, then the diophantine equation x 2 þ q 2kþ1 ¼ y n has exactly two families of solutions ðq; n; k; x; yÞ.

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Let p # [3, 23] and D # N such that p |3 D and (D, p){(2, 3). We prove in this paper that the diophantine equation x, z # N has at most one solution (x, z). Moreover, we give an explicit upper bound for z.