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

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

2003-01-24

arxiv.org/abs/math/0301274v3

Fundamenta Informaticae 56 (2003) 273--284

Mathematics

Number Theory

Scientific paper

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.

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