Some Remarks on the Diophantine Equations x2 − Dy4 = 1 and x4 − Dy2 = 1
Some Remarks on the Diophantine Equations x2 − Dy4 = 1 and x4 − Dy2 = 1
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DOI：
10.1112/jlms/s1-41.1.542
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发表时间：
1966
影响因子：
1.2
通讯作者：
W. Ljunggren
中科院分区：
文献类型：
--
作者：
W. Ljunggren
In two previous papers [1] and [2] I have proved that the diophantine equations x2-@ y*=\(1) and x*-@ y2= l(2) where Q>> 0 and is not a perfect square, has at most two solutions in positive integers, and that these can be found when the fundamental unit e is known in the quadratic field Q {\/Q)). Recently Professor LJ Mordell [3] has published a theorem concerning explicit formulae for Q) for which (1) has no solutions in positive integers. As a special result he found that (1) has no such solutions if 2 is an odd prime, 3>= 5, 9 or 13 (mod 16), except@= 5 where {x, y)=(3, 2) is the only solution. Mordell says that he cannot deal with the case Si= 1 (mod 16). The main purpose of this note is to show that Mordell's result is valid for all primes p~ l (mod 4). Besides I add a similar theorem for the equation (2). In both cases we make use of the theorem that the only solutions in positive integers of are (x, y)=(1, 1),(13, 239). See [4], It is well-known that N (e)=—1 in case S)= p= 1 (mod 4). Then (1) implies
