The Erdős-Straus Conjecture

Here’s a cute little problem.

Can every number of the form 4/n (where n is an integer) be written as the sum of three unit fractions? (A unit fraction is a fraction of the form 1/a where a is an integer.)

For example, if n = 5 we have

\frac{4}{5} = \frac{1}{2} + \frac{1}{4} + \frac{1}{20}.

There’s been plenty of work on this conjecture, originally proposed by the prolific Paul Erdős. It’s known that the conjecture holds true for certain arithmetic progressions. Furthermore, we only have to prove the statement for when n is a prime. Well, the primes can be divided into two sets as to whether they are congruent to 1 or 2 \mod 3. For primes p of the latter type, we have that

\frac{4}{p} = \frac{1}{p} + \frac{3}{p+1} + \frac{3}{p(p+1)}.

 

How tantalising! Of course with number theory being the devious little minx that she is, this approach does not seem possible for primes which are 1 more than a multiple of 3. What to do here? We could split this congruence class into more classes and keep attacking it this way, though you’re bound to get tired at some point. But it’s still good clean fun!

Yes, I’ve tried to work on this conjecture using elementary methods. I think it’s a good idea to try the problems which are known to be difficult. This way, you get a good “feel” for the problem and what approaches might work.

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