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Simultaneous rational approximations to certain algebraic numbers

Published online by Cambridge University Press:  24 October 2008

A. Baker
A. Baker
Affiliation:
Trinity College, Cambridge
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Extract

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the form

is satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, 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, …, αk 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.‡

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 3 , July 1967 , pp. 693 - 702
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

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