1999/03/22 by Greg Martin · 2 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Limits and Structures in Graph Theory #Annotation #Algorithm #Type (biology) #Semantics (computer science) #Mathematics #Computer science #Artificial intelligence #Biology #Programming language
paper · pdf · doi:10.1090/s0002-9947-99-02327-2
openalex publication_date 1999/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest denominator. We show that for every positive rational there exist representations as Egyptian fractions whose largest denominator is at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and whose denominators form a positive proportion of the integers up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for sufficiently large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; furthermore, the proportion is within a small factor of best possible.