The equation of Ramanujan-Nagell and [𝑦²]

Abstract

By arithmetizing Levi’s constructive test for membership in [<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket y squared right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[{y^2}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>] we have translated the questions of whether a given power product is in [<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket y squared right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[{y^2}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>] to determining whether a certain product of matrices is the zero matrix. This leads to number-theoretic problems, including the diophantine equations of the title <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n Baseline minus 7 equals x squared"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>7</mml:mn> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^n} - 7 = {x^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Proceedings of the American Mathematical Society - View - PDF