The
ErdösStrauss conjecture
Introduction. The greedyalgorithm result tells one that

where
is a fraction whose numerator
is
less
than
a
.

where
is a fraction whose numerator
is
less
than

where
is a fraction whose numerator
is
less
than
etc
The sequence
is
strictly
monotonic decreasing, which eventually terminates when one has
, for some
r
. For a given
a
, there will then be at most
a
terms in the sequence of unit fractions:
, and clearly the maximum can happen in the event that the sequence
happens to be the natural numbers less than
a
:
.
Thus,
using the greedy algorithm
:

will require at most 2 unit fractions

will require at most 3 unit fractions

will require at most 4 unit fractions

will require at most 5 unit fractions
etc
Conjecture (ErdösStrauss). For every natural number
n
, greater than 3, there is an Egyptian fraction representation that is better than the one ensured by the greedy algorithm representation; in other words: for every integer
n
(greater than 3) there are natural numbers
x
,
y
and
z
such that
Interested readers may consult Chapter 30 of Mordell's
Diophantine Equations
.
Some elementary ideas (which could be at school level). Using
Thus, setting
n
= 3
x
+ 2 (
x
= 0, 1, 2, 3, 4, 5, ... ), we have that the ErdösStrauss conjecture is (trivially) true for all such
n
.