2016/10/19 by William B. Hart, David Harvey, Wilson Ong · 18 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #Mathematics #Conjecture #Constant (computer programming) #Class (philosophy) #Combinatorics #Discrete mathematics #Arithmetic #Pure mathematics
paper · pdf · doi:10.1090/mcom/3211
published in Mathematics of Computation 86(308), 3031-3049 (American Mathematical Society)
openalex publication_date 2016/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We compute all irregular primes less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript 31 Baseline equals 2 147 483 648"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>31</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mn>147</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mn>483</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mn>648</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">231 = 2 147 483 648</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We verify the Kummer–Vandiver conjecture for each of these primes, and we check that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -part of the class group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q left-parenthesis zeta Subscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> ζ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf Q(ζ p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the simplest possible structure consistent with the index of irregularity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our method for computing the irregular indices saves a constant factor in time relative to previous methods, by adapting Rader’s algorithm for evaluating discrete Fourier transforms.