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Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.3/1006
- Title
- On the existence of a new family of Diophantine equations for Ω [Omega]
- Author(s)
- Ord, Toby; Kieu, Tien D.
- Abstract
- We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω [Omega] by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω [Omega] 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 [Omega] 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 [Omega] is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values iff the k-th bit of Ω [Omega] is 1.
- Publication type
- Journal article
- Research centre
- Swinburne University of Technology. Centre for Atom Optics and Ultrafast Spectroscopy
- Source
- Fundamenta Informaticae, Vol. 56, no. 3 (Aug. 2003), pp. 273-284
- Publication year
- 2003
- Keyword(s)
- Adiophantine equation; Algorithmic information theory; Hilbert's tenth problem; Information theory; Omega; Randomness; Random sets
- Publisher
- IOS Press
- ISSN
- 0169-2968
- Copyright
- Copyright © 2003 IOS Press.
- Peer reviewed
