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The Diophantine Equation 3u2−2=v6

✍ Scribed by Harun P.K. Adongo

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
312 KB
Volume
59
Category
Article
ISSN
0022-314X

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✦ Synopsis

The theorem of Delaunay Nagell states that: If d is a cube-free integer >1, then the equation x 3 +dy 3 =1 has at most one solution in non-zero integers x, y, and if such a solution exists then x+ y 3 -d is either the fundamental unit of the field Q( 3d ) or its square, the latter occurring for only finitely many values of d. Investigation of these exceptional d values has led to the equation of the title [5, 3.9], which has only finitely many solutions. We prove that the title equation has no integer solution other than |u|=|v| =1, which give the known values d=19, 20, 28, therefore there are no other d values.

1996 Academic Press, Inc. (i) Is a Dedekind domain (ii) The set [1, -6] forms an integral basis (iii) Is a PID, hence UFD (iv) The group U of units is isomorphic to [(5+2 -6) k | k # Z].

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