A note on monte carlo primality tests and algorithmic information theory

Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose conclusions are only probably correct. On the other hand, algorithmic information theory provides a precise mathematical definition of the notion of random or patternless sequence. In this paper we shall describe conditions under which if the sequence of coin tosses in the Solovay-Strassen and Miller-Rabin algorithms is replaced by a sequence of heads and tails that is of maximal algorithmic information content, i.e., has maximal algorithmic randomness, then one obtains an error-free test for primality. These results are only of theoretical interest, since it is a manifestation of the Gödel incompleteness phenomenon that it is impossible to certify a sequence to be random by means of a proof, even though most sequences have this property. Thus by using certified random sequences one can in principle, but not in practice, convert proba...