Simultaneous rational approximations to certain algebraic numbers

Publisher Website
Google Scholar
Abstract
It is generally conjectured that if α1, α2 …, α k are algebraic numbers for which no equation of the form is satisfied with rational r i not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p 1, p 2, …, p k q, q > 0, such that This result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, α k are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

This publication has 18 references indexed in Scilit: