On a Quartic Diophantine Equation
نویسندگان
چکیده
The wish to determine the complete set of rational integral solutions of (1) was expressed by Diaconis and Graham in [2, p. 328]. Apparently, the solutions to this diophantine problem correspond to values of keN for which the Radon transform based on the set of all xeZ\ with exactly four ones is not invertible. We thank Hendrik Lenstra who communicated the problem to Jaap Top, to whom we are equally grateful for pointing it out to us. In the sequel we shall solve equation (1) completely by reducing this equation to a finite set of Thue equations, which are subsequently dealt with individually. In fact we have to solve five different quartic Thue equations. We feel the original equation to be sufficiently interesting to warrant a twofold solution process, emphasizing algebraic as well as diophantine approximation techniques. The diophantine approximation approach rests on the theory of linear forms in the logarithms of algebraic numbers and follows the lines set out in [7] and [6], and the algebraic approach is based on the properties of certain binary sequences with values in a quadratic subfield of the biquadratic number field associated with the relevant Thue equation (see [3]).
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