Explicit Bounds for Primality Testing and Related Problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is <!-- MATH $O({\log ^2}m)$ --> . This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most <!-- MATH $2{\log ^2}m$ --> .