The Erdős–Straus conjecture is an
unproven statement in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. The conjecture is that, for every
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
that is greater than or equal to 2, there exist positive integers
,
, and
for which
In other words, the number
can be written as a sum of three positive
unit fraction
A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When a ...
s.
The conjecture is named after
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
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 equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s.
Although a solution is not known for all values of , infinitely many values in certain infinite
arithmetic progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
s have simple formulas for their solution, and skipping these known values can speed up searches for
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
s. Additionally, these searches need only consider values of
that are
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, because any composite counterexample would have a smaller counterexample among its
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. Computer searches have verified the truth of the conjecture up to
.
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.
Background and history
When a rational number is expanded into a sum of unit fractions, the expansion is called an
Egyptian fraction. This way of writing fractions dates to the
mathematics of ancient Egypt, in which fractions were written this way instead of in the more modern
vulgar fraction form
with a numerator
and denominator
. The Egyptians produced tables of Egyptian fractions for unit fractions multiplied by two, the numbers that in modern notation would be written
, such as the
Rhind Mathematical Papyrus table; in these tables, most of these expansions use either two or three terms. These tables were needed, because the obvious expansion
was not allowed: the Egyptians required all of the fractions in an Egyptian fraction to be different from each other. This same requirement, that all fractions be different, is sometimes imposed in the Erdős–Straus conjecture, but it makes no significant difference to the problem, because for
any solution to
where the unit fractions are not distinct can be converted into a solution where they are all distinct;
see below.
[
Although the Egyptians did not always find expansions using as few terms as possible, later mathematicians have been interested in the question of how few terms are needed. Every fraction has an expansion of at most terms, so in particular needs at most two terms, needs at most three terms, and needs at most four terms. For , two terms are always needed, and for , three terms are sometimes needed, so for both of these numerators, the maximum number of terms that might be needed is known. However, for , it is unknown whether four terms are sometimes needed, or whether it is possible to express all fractions of the form using only three unit fractions; this is the Erdős–Straus conjecture. Thus, the conjecture covers the first unknown case of a more general question, the problem of finding for all the maximum number of terms needed in expansions for fractions .
One way to find short (but not always shortest) expansions uses the ]greedy algorithm for Egyptian fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a s ...
, first described in 1202 by Fibonacci
Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
The name he is commonly called, ''Fibonacci ...
in his book ''Liber Abaci
The or (Latin for "The Book of Calculation") was a 1202 Latin work on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. It is primarily famous for introducing both base-10 positional notation and the symbols known as Arabic n ...
''. This method chooses one unit fraction at a time, at each step choosing the largest possible unit fraction that would not cause the expanded sum to exceed the target number. After each step, the numerator of the fraction that still remains to be expanded decreases, so the total number of steps can never exceed the starting numerator, but sometimes it is smaller. For example, when it is applied to , the greedy algorithm will use two terms whenever is 2 modulo 3, but there exists a two-term expansion whenever has a factor that is 2 modulo 3, a weaker condition. For numbers of the form , the greedy algorithm will produce a four-term expansion whenever is 1 modulo 4, and an expansion with fewer terms otherwise. Thus, another way of rephrasing the Erdős–Straus conjecture asks whether there exists another method for producing Egyptian fractions, using a smaller maximum number of terms for the numbers .
The Erdős–Straus conjecture was formulated in 1948 by Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
and Ernst G. Straus, and published by . Richard Obláth also published an early work on the conjecture, a paper written in 1948 and published in 1950, in which he extended earlier calculations of Straus and Harold N. Shapiro in order to verify the conjecture for all .
Formulation
The conjecture states that, for every integer , there exist positive integers , , and such that
For instance, for , there are two solutions:
Multiplying both sides of the equation by leads to an equivalent polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
form for the problem.
Distinct unit fractions
Some researchers additionally require that the integers , , and be distinct from each other, as the Egyptians would have, while others allow them to be equal. For , it does not matter whether they are required to be distinct: if there exists a solution with any three integers, then there exists a solution with distinct integers.
conflict resolution section
This is because two identical unit fractions can be replaced through one of the following two expansions:
(according to whether the repeated fraction has an even or odd denominator) and this replacement can be repeated until no duplicate fractions remain. For , however, the only solutions are permutations of .
Negative-number solutions
The Erdős–Straus conjecture requires that all three of , , and be positive. This requirement is essential to the difficulty of the problem. Even without this relaxation, the Erdős–Straus conjecture is difficult only for odd values of , and if negative values were allowed then the problem could be solved for every odd by the following formula:
Computational results
If the conjecture is false, it could be proven false simply by finding a number that has no three-term representation. In order to check this, various authors have performed brute-force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of Iteration#Computing, systematically checking all possible candida ...
es for counterexamples to the conjecture. Searches of this type have confirmed that the conjecture is true for all up to .
In such searches, it is only necessary to look for expansions for numbers where is a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. This is because, whenever has a three-term expansion, so does for all positive integers . To find a solution for , just divide all of the unit fractions in the solution for by :
If were a counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
to the conjecture, for a composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
, every prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of would also provide a counterexample that would have been found earlier by the brute-force search. Therefore, checking the existence of a solution for composite numbers is redundant, and can be skipped by the search. Additionally, the known modular identities for the conjecture (see below) can speed these searches by skipping over other values known to have a solution. For instance, the greedy algorithm finds an expansion with three or fewer terms for every number where is not 1 modulo 4, so the searches only need to test values that are 1 modulo 4. One way to make progress on this problem is to collect more modular identities, allowing computer searches to reach higher limits with fewer tests.
The number of distinct solutions to the problem, as a function of , has also been found by computer searches for small and appears to grow somewhat irregularly with . Starting with , the numbers of distinct solutions with distinct denominators are
Even for larger there can sometimes be relatively few solutions; for instance there are only seven distinct solutions for .
Theoretical results
In the form , a polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
with integer variables, the Erdős–Straus conjecture is an example of a Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
. The Hasse principle for Diophantine equations suggests that these equations should be studied using modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
. If a polynomial equation has a solution in the integers, then taking this solution modulo , for any integer , provides a solution in modulo- arithmetic. In the other direction, if an equation has a solution modulo for every prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...
, then in some cases it is possible to piece together these modular solutions, using methods related to the Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
, to get a solution in the integers. The power of the Hasse principle to solve some problems is limited by the Manin obstruction, but for the Erdős–Straus conjecture this obstruction does not exist.
On the face of it this principle makes little sense for the Erdős–Straus conjecture. For every , the equation is easily solvable modulo any prime, or prime power, but there appears to be no way to piece those solutions together to get a positive integer solution to the equation. Nevertheless, modular arithmetic, and identities based on modular arithmetic, have proven a very important tool in the study of the conjecture.
Modular identities
For values of satisfying certain congruence relations, one can find an expansion for automatically as an instance of a polynomial identity. For instance, whenever is 2 modulo 3, has the expansion
Here each of the three denominators , , and is a polynomial of , and each is an integer whenever is 2 modulo 3. The greedy algorithm for Egyptian fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a s ...
finds a solution in three or fewer terms whenever is not 1 or 17 mod 24, and the 17 mod 24 case is covered by the 2 mod 3 relation, so the only values of for which these two methods do not find expansions in three or fewer terms are those congruent to 1 mod 24.
Polynomial identities listed by provide three-term Egyptian fractions for whenever is one of:
*2 mod 3 (above),
*3 mod 4,
*2 or 3 mod 5,
*3, 5, or 6 mod 7, or
*5 mod 8.
Combinations of Mordell's identities can be used to expand for all except possibly those that are 1, 121, 169, 289, 361, or 529 mod 840. The smallest prime that these identities do not cover is 1009. By combining larger classes of modular identities, Webb and others showed that the natural density of potential counterexamples to the conjecture is zero: as a parameter goes to infinity, the fraction of values in the interval