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• W2017738613 abstract "We have determined the irregular primes below 125000 and tabulated their distribution. Two primes of index five of irregularity were found, namely 78233 and 94693. Fermat’s Last Theorem has been verified for all exponents up to 125000. We computed the cyclotomic invariants <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu Subscript p> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{mu _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda Subscript p> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{lambda _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=nu Subscript p> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>ν<!-- ν --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{nu _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and found that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu Subscript p Baseline equals 0> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>{mu _p} = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 125000> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>125000</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p > 125000</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The complete factorizations of the numerators of the Bernoulli numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B Subscript 2 k> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{B_{2k}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 k less-than-or-slanted-equals 60> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mn>60</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>2k leqslant 60</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Euler numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript 2 k> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{E_{2k}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 k less-than-or-slanted-equals 42> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mn>42</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>2k leqslant 42</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given." @default.
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• W2017738613 date "1978-01-01" @default.
• W2017738613 modified "2023-09-29" @default.
• W2017738613 title "The irregular primes to 125000" @default.
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