A quadratic partition of primes ≡1 (𝑚𝑜𝑑7)

Abstract

The solutions of a quadratic partition of primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p identical-to 1 left-parenthesis mod 7 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.667em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em" /> <mml:mn>7</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p \equiv 1 \pmod 7</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in terms of which the author and P. A. Leonard have given the cyclotomic numbers of order seven and also necessary and sufficient conditions for 2, 3, 5 and 7 to be seventh powers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis mod p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mspace width="0.667em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em" /> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pmod p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, are obtained for all such primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than 1000"> <mml:semantics> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1000</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">&gt; 1000</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Mathematics of Computation - View - PDF