Erdős–Straus conjecture
E554304
The Erdős–Straus conjecture is an unsolved problem in number theory asserting that for every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | mathematical conjecture ⓘ |
| concerns |
Egyptian fractions
NERFINISHED
ⓘ
representation of rational numbers as sums of unit fractions ⓘ |
| difficulty | considered hard in elementary number theory ⓘ |
| domainOfVariable | positive integers n ≥ 2 ⓘ |
| field | number theory ⓘ |
| formalStatement | For every integer n ≥ 2, there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. ⓘ |
| hasAlternativeFormulation | For every n ≥ 2, the Diophantine equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers x, y, z. ⓘ |
| hasComputationalVerification | verified for all n up to very large bounds by computer search ⓘ |
| hasForm | 4/n = 1/x + 1/y + 1/z with x, y, z ∈ ℕ ⓘ |
| hasNotation | 4/n = 1/a + 1/b + 1/c ⓘ |
| hasPartialResults |
proved for all n below large explicit bounds
ⓘ
proved for all n in many congruence classes modulo various integers ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| impliedBy | truth for all prime n ≥ 2 ⓘ |
| influenced | research on Egyptian fractions ⓘ |
| involvesEquation | 4nxyz = nxy + nxz + nyz ⓘ |
| isSpecialCaseOf | problems on expressing rational numbers as sums of few unit fractions ⓘ |
| languageOfOriginalPublication | English ⓘ |
| literatureType | research papers in analytic and elementary number theory ⓘ |
| motivation | understanding structure of Egyptian fraction representations ⓘ |
| namedAfter |
Ernst G. Straus
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| namedBy | Paul Erdős and Ernst G. Straus NERFINISHED ⓘ |
| openProblemIn |
Diophantine equations
ⓘ
elementary number theory ⓘ |
| reduction | It suffices to prove the conjecture for prime n. ⓘ |
| relatedConcept |
Egyptian fraction decomposition
NERFINISHED
ⓘ
unit fraction ⓘ |
| relatedConjecture | Egyption fraction conjectures NERFINISHED ⓘ |
| statement | For every integer n ≥ 2, the fraction 4/n can be expressed as a sum of three unit fractions. ⓘ |
| status | open ⓘ |
| subfield |
Diophantine analysis
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ⓘ
additive number theory ⓘ |
| topic | representation of 4/n as sum of three unit fractions ⓘ |
| typicalMethod |
case analysis by congruence classes of n
ⓘ
computer-assisted search for representations ⓘ |
| unknownFor | all n in general ⓘ |
| usesConcept |
Diophantine equations
NERFINISHED
ⓘ
computational number theory ⓘ modular arithmetic ⓘ |
| variableConstraint |
n is an integer with n ≥ 2
ⓘ
x, y, z are positive integers ⓘ |
| yearProposed | 1948 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.