This is a **famous unsolved problem in number theory**.
### Short answer
* **It is conjectured that YES**, for **every integer ( n \ge 2 )** there exist **positive integers** ( x,y,z ) such that
[
\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
]
* This statement is called the **Erd?s�Straus Conjecture** (1948).
* **It has not been proven in full generality**, but it has been **verified for all ( n ) up to extremely large bounds** (well beyond (10^{17})) and proven for many infinite classes of ( n ).
—
## What *is* known for sure
### ? Verified computationally
The equation has been checked and **works for every ( n \ge 2 )** up to enormous limits by computer search.
### ? Proven for many infinite families
For example, solutions are known when:
* ( n \equiv 0,1,2 \pmod{4} )
* ( n ) has certain prime factorizations
* ( n ) is even
* ( n ) is divisible by many primes
Only certain rare forms of odd numbers require deeper checking�but all tested cases work.
—
## Example solutions
### Example 1: ( n = 2 )
[
\frac{4}{2} = 2 = \frac{1}{1} + \frac{1}{1} + \frac{1}{\infty}
]
(Trivial case; usually excluded.)
### Example 2: ( n = 3 )
[
\frac{4}{3} = \frac{1}{1} + \frac{1}{4} + \frac{1}{12}
]
### Example 3: ( n = 5 )
[
\frac{4}{5} = \frac{1}{2} + \frac{1}{4} + \frac{1}{20}
]
### Example 4: General construction (one of many)
If ( n \equiv 2 \pmod{4} ), then:
[
\frac{4}{n}
= \frac{1}{\frac{n}{2}}
* \frac{1}{n}
* \frac{1}{n}
]
—
## Why this is hard
* Writing a rational number as a sum of **unit fractions** (Egyptian fractions) is highly non-unique.
* The difficulty is proving **existence for all integers**, not finding examples.
* No counterexample is known.
—
## Final conclusion
* ?? **Yes, it is believed that such a decomposition always exists**
* ? **But it has not been proven for all ( n \ge 2 )**
* ? This is a major open conjecture in mathematics