2026/07/21 by Carl Pomerance, CARL POMERANCE
Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #Advanced Algebra and Geometry
paper · pdf · doi:10.1017/s0004972726101610
Abstract For each pair <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:math> j,n j comma n with <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mi>j</mml:mi> </mml:math> n>j n greater than j , we consider the subgroup of the multiplicative group mod n of residues with order dividing <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>j</mml:mi> </mml:math> n-j n minus j . We generalise some results in the case <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> j=1 j equals 1 due to Erdős and the current author [‘On the number of false witnesses for a composite number’, Math. Comp. 46 (1986), 259–279]. The case <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> j=0 j equals 0 is of particular interest.