Erdős–Straus conjecture

The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer

n

{\displaystyle n}

that is greater than or equal to 2, there exist positive integers

x

{\displaystyle x}

,

y

{\displaystyle y}

, and

z

{\displaystyle z}

for which

4

n

=

1

x

+

1

y

+

1

z

.

{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}.}

In other words, the number

4

/

n

{\displaystyle 4/n}

can be written as a sum of three positive unit fractions.

The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations.

Although a solution is not known for all values of n, infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values can speed up searches for counterexamples. Additionally, these searches need only consider values of

n

{\displaystyle n}

that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified the truth of the conjecture up to

n

≤

10

17

{\displaystyle n\leq 10^{17}}

.

If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied.