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On the existence of a new family of Diophantine equations for $\bf Ω$

Toby Ord·Tien D. Kieu
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Abstract

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.

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Toby Ord, Tien D. Kieu. 2003-10-12. On the existence of a new family of Diophantine equations for $\bf Ω$. https://arxiv.org/abs/math/0301274

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