Represent improper fraction as a sum of unique unit fractions Is it possible to represent an improper fraction as a finite sum of unique unit fractions (Egyptian fractions)?

Dylan Nixon

Dylan Nixon

Answered question

2022-10-25

Represent improper fraction as a sum of unique unit fractions
Is it possible to represent an improper fraction as a finite sum of unique unit fractions (Egyptian fractions)?

Answer & Explanation

Cavalascamq

Cavalascamq

Beginner2022-10-26Added 21 answers

Yes. Start with α Q , α > 1. Then let n be the greatest integer such that
H n = i = 1 n 1 i < α
Of course n exists because the infinite Harmonic series diverges.
It follows that α H n < 1 n so none of the fractions in the standard Egyptian decomposition of α H n can appear in H n

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