Algorithmic information theory
Algorithmic information theory
Hilbert's tenth problem
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
A characterization of c.e. random reals
Theoretical Computer Science
Minds and Machines
The Halting Probability via Wang Tiles
Fundamenta Informaticae
An Algebraic Characterization of the Halting Probability
Fundamenta Informaticae
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We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω by 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 iff the k-th bit of Ω is 1.