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A note on integral integer-valued functions of several variables

Published online by Cambridge University Press:  24 October 2008

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

1. The consequences of assuming that an integral function or its derivatives take integer values for some or all integer values of the variable have been widely investigated. The best-known result of this kind was obtained by Pólya(9,10) and Hardy(6) in 1920 and states that among all transcendental integral functions which assume integer values for all non-negative integer values of the variable that of least increase is the function 2Z (cf. also (7)). Analogous results relating to integers in the Gaussian field were proved by Fukasawa(3)in 1926 and Gelfond(4)in 1929, and theorems relating to the function and its derivatives are included in the works of Gelfond (5), Selberg(14) and Straus (15). (For further results and references see (1,2, 11).) The earlier of these investigations may be regarded as the genesis of the celebrated Gelfond-Schneidersolution to the seventh problem of Hilbert and the much important work arising therefrom.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 3 , July 1967 , pp. 715 - 720
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

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