On the existence of a new family of Diophantine equations for Ω

Authors:
Toby Ord;Tien D. Kieu
Affiliations:
Department of Philosophy, The University of Melbourne, Parkville 3010, Australia;Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Hawthorn 3122, Australia
Venue:
Fundamenta Informaticae
Year:
2002

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Abstract

We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, has solutions for an odd or an even number of values of N. We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter k which has an odd number of solutions iff the k-th bit of Ω is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values if the k-th bit of Ω is 1.