Abstract
The greatest common divisor of two integers cannot be generated in a uniformly bounded number of steps from those integers using arithmetic operations. The proof uses an elementary model-theoretic construction that enables us to focus on “integers with transcendental ratio.” This unboundedness result is part of the solution of a problem posed by Y. Moschovakis on limitations of primitive recursive algorithms for computing the greatest common divisor function.
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References
C. Chang and J. Keisler, Model Theory, 3rd ed., North-Holland, Amsterdam, 1990
P. Cijsouw (1974) ArticleTitleTranscendence measures of exponentials and logarithms of algebraic numbers Compositio Math. 28 163–178 Occurrence Handle0284.10013
L. Colson (1991) ArticleTitleAbout primitive recursive algorithms Theoret. Comput. Sci. 83 57–69 Occurrence Handle0744.03042
L. van den Dries and A.J. Wilkie, The laws of integer divisibility, and solution sets of linear divisibility conditions, J. Symbolic Logic (to appear)
W. LeVeque, Fundamentals of Number Theory, Addison-Wesley, Reading, MA, 1977
Y. Moschovakis, On primitive recursive algorithms and the greatest common divisor function, {Theoret. Comput. Sci. (to appear)
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Communicated by Peter Olver
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van den Dries, L. Generating the Greatest Common Divisor, and Limitations of Primitive Recursive Algorithms. Found Comput Math 3, 297–324 (2003). https://doi.org/10.1007/s10208-002-0061-y
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DOI: https://doi.org/10.1007/s10208-002-0061-y