Explicit bounds for primality testing and related problems

Abstract

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 <italic>m</italic> omits a number that is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis log squared m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({\log ^2}m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <italic>m</italic> has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 log squared m"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2{\log ^2}m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.