A note on monte carlo primality tests and algorithmic information theory
Communications on Pure and Applied Mathematics
Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose con- G. J. Chaitin clusions 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
... 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 probabilistic tests for primality into deterministic ones. 1 The second author has been supported by US DOE, Contract EY-76-C-02-3077*000. We wish to thank John Gill III and Charles Bennett for helpful discussions. Reproduction in whole or in part is permitted for any purpose of the United States Government. 2 We could equally well have used in this paper the newer formalism of , in which programs are "self-delimiting."