On the existence of a new family of Diophantine equations for $\bf \Omega$
<|reference_start|>On the existence of a new family of Diophantine equations for $\bf \Omega$: We show how to determine the $k$-th bit of Chaitin's algorithmically random real number $\Omega$ by solving $k$ instances of the halting problem. From this we then reduce the problem of determining the $k$-th bit of $\Omega$ 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 $\Omega$ in number theory: an exponential Diophantine equation with a parameter $k$ which has an odd number of solutions iff the $k$-th bit of $\Omega$ 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 $\Omega$ is 1.<|reference_end|>
@article{ord2003on,
    title={On the existence of a new family of Diophantine equations for $\bf
  \Omega$},
    author={Toby Ord and Tien D. Kieu},
    journal={Fundamenta Informaticae 56 (2003) 273--284},
    year={2003},
    archivePrefix={arXiv},
    eprint={math/0301274},
    primaryClass={math.NT cs.CC quant-ph}
}