On the existence of a new family of Diophantine equations for z

We show how to determine the -th bit of Chaitin’s algorithmically random real number by solving instances of the halting problem. From this we then reduce the problem of determining the -th bit of to determining whether a certain Diophantine equation with two parameters, and , has solutions for an odd or an even number of values of . We also demonstrate two further examples of in number theory: an exponential Diophantine equation with a parameter which has an odd number of solutions iff the -th bit of is 1, and a polynomial of positive integer variables and a parameter that takes on an odd number of positive values iff the -th bit of is 1.