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...

The main thrust of this article is to show how complete solutions of quadratic Diophantine equations can be given, for any positive discriminant, in terms of the continued fraction algorithm. This is in response to recent results by Zhang [4]-[6], wherein semi-simple continued fractions were introduced to generalize the well-known fact that solutions of quadratic Diophantine equations with valu...