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W2019924527 abstract "Wendt’s binomial circulant determinant,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W Subscript m><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:msub><mml:mi>W</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>{W_m}</mml:annotation></mml:semantics></mml:math></inline-formula>, is the determinant of an<italic>m</italic>by<italic>m</italic>circulant matrix of integers, with (<italic>i, j</italic>)th entry<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartBinomialOrMatrix m Choose StartAbsoluteValue i minus j EndAbsoluteValue EndBinomialOrMatrix><mml:semantics><mml:mrow><mml:mo>(</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mtable rowspacing=4pt columnspacing=1em><mml:mtr><mml:mtd><mml:mi>m</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>|</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>left ( {begin {array}{*{20}{c}} m {|i - j|} end {array} } right )</mml:annotation></mml:semantics></mml:math></inline-formula>whenever 2 divides<italic>m</italic>but 3 does not. We explain how we found the prime factors of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W Subscript m><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:msub><mml:mi>W</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>{W_m}</mml:annotation></mml:semantics></mml:math></inline-formula>for each even<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m less-than-or-equal-to 200><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:mn>200</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>m leq 200</mml:annotation></mml:semantics></mml:math></inline-formula>by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if<italic>p</italic>and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q equals m p plus 1><mml:semantics><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mi>m</mml:mi><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>q = mp + 1</mml:annotation></mml:semantics></mml:math></inline-formula>are odd primes, 3 does not divide<italic>m</italic>, and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m less-than-or-equal-to 200><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:mn>200</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>m leq 200</mml:annotation></mml:semantics></mml:math></inline-formula>, then the first case of Fermat’s Last Theorem is true for exponent<italic>p</italic>." @default.
W2019924527 created "2016-06-24" @default.
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W2019924527 date "1991-01-01" @default.
W2019924527 modified "2023-09-29" @default.
W2019924527 title "The prime factors of Wendt’s binomial circulant determinant" @default.
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