Problems & Puzzles:
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
"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 , 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  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
By his side, Emmanuel Vantieghem found a
solution for the same problem but using
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
Special request: Please send your
results to both questions in a text file just one denominator per
Contribution came from Emmanuel Vantieghem.
Emmanuel wrote, around November 3, 2017:
Here is the annex.
You can find a 73035 solution
for Q2 in annex. I 'm almost sure that there exist smaller
I worked every evening of this week on finding an answer to Q1,
On Set 8, 2010, Tatsuru Watanabe wrote:
We found 17 results for Q1 each of which has 47 terms.
One example is this:
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