Problems & Puzzles: Puzzles

Puzzle 899. A follow-up to Puzzle 897 In Puzzle 897 we shown a result by A. W. Johnson, dated on 1978: an Egyptian fraction equal to 1, composed by 48 terms. In the publication where Johnson gave his results, he wrote: "In this journal T1977: 178-181], E.J. Barbeau treats the representation of one as a sum of reciprocals of distinct positive integers, each of which is the product of exactly two primes. He exhibits a set containing 101 such integers and asks what are the fewest integers required in a representation of this type. As reported in [1], at least 38 integers are required in the smallest set, and sets containing as few as 50 integers are known to exist. I discovered too late to be included in [1] the following set of 48 integers of the required type" Q1. Can you discover a result of Egyptian fraction equal to 1, with less than 48 terms each of which is the product of exactly two primes? By his side, Emmanuel Vantieghem found a solution for the same problem but using 73082 terms each of which is the product of exactly three primes (see Q3 for Puzzle 897). One day after publishing his results in my pages, I asked to Emmanuel, "would you say that your solution with 73082 terms to Q3, is the minimal one regarding the quantity of terms". His answer was "I guess It's not minimal". Q2. Can you discover a result of Egyptian fraction equal to 1, with less than 73082 terms each of which is the product of exactly three primes? Special request: Please send your results to both questions in a text file just one denominator per row.

---

Contribution came from Emmanuel Vantieghem.

***

Emmanuel wrote, around November 3, 2017:

You can find a 73035  solution for  Q2  in annex. I 'm almost sure that there exist smaller sets.

I worked every evening of this week on finding an answer to Q1, without success.

Here is the annex.

***

On Set 8, 2010, Tatsuru Watanabe wrote:

We found 17 results for Q1 each of which has 47 terms.

One example is this:

 

{6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62
65,69,74,82,85,86,87,91,93,95,106,111,123,133,145
155,159,185,203,215,253,265,287,319,493,583,731,851
1073}
 

It seems that there are no examples that have less than 47 terms.

If you want to know more details, please read this paper or rear it here

https://arxiv.org/pdf/2009.03275.pdf.

***

 

 

---