1999/05/17 by Douglas E. Iannucci · 5 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Rings, Modules, and Algebras
paper · pdf · doi:10.1090/s0025-5718-99-01126-6
openalex publication_date 1999/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">σ (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the sum of positive divisors of the natural number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Such a number is said to be <italic>perfect</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis n right-parenthesis equals 2 n"> <mml:semantics> <mml:mrow> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">σ (n)=2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is well known that a number is even and perfect if and only if it has the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript p minus 1 Baseline left-parenthesis 2 Superscript p Baseline minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2p-1 (2p-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript p Baseline minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2p-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 Superscript 6"> <mml:semantics> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">106</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and Hagis showed that the second largest must exceed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 cubed"> <mml:semantics> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">103</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we improve the latter bound. In particular, we prove the statement in the title of this paper.