Erdős–Moser equation
E554305
The Erdős–Moser equation is a famous unsolved Diophantine equation in number theory that asks whether 1^k + 2^k + ... + (m−1)^k = m^k has any integer solutions beyond the trivial case (k, m) = (1, 2).
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
Diophantine equation
ⓘ
unsolved problem in number theory ⓘ |
| appearsIn | research literature on Diophantine equations ⓘ |
| asksWhether | there exist integer solutions (k,m) with k>1 and m>2 ⓘ |
| conjecturesThat | no solutions exist with k>1 ⓘ |
| difficulty | considered very hard ⓘ |
| domainOfVariables | positive integers ⓘ |
| field |
Diophantine analysis
ⓘ
number theory ⓘ |
| hasConjectureFormulation | if 1^k + 2^k + ··· + (m−1)^k = m^k with integers k,m≥2 then no such pair exists ⓘ |
| hasForm | 1^k + 2^k + ··· + (m−1)^k = m^k ⓘ |
| hasTrivialCase |
k=1 gives 1 = 2^1 − 1
ⓘ
m=2 gives 1^k = 2^k for k=1 only ⓘ |
| hasTrivialSolution | (k,m) = (1,2) ⓘ |
| involvesOperation |
finite sums
ⓘ
integer powers ⓘ |
| isRelatedTo |
Erdős–Moser conjecture
NERFINISHED
ⓘ
Prouhet–Tarry–Escott problem NERFINISHED ⓘ Waring-type problems ⓘ equal sums of like powers ⓘ perfect powers ⓘ sum of powers function ⓘ |
| isSpecialCaseOf | power sum equations ⓘ |
| knownResult |
(k,m) = (1,2) is the only solution with k ≤ 1
ⓘ
no solutions are known with k>1 ⓘ |
| namedAfter |
Leo Moser
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| openQuestion | whether any nontrivial integer solutions exist ⓘ |
| proposedBy | Leo Moser NERFINISHED ⓘ |
| researchTheme |
bounding possible values of m and k
ⓘ
use of analytic number theory methods ⓘ use of computational searches for solutions ⓘ |
| solutionConstraint |
k is a positive integer
ⓘ
k ≥ 1 ⓘ m is a positive integer ⓘ m ≥ 2 ⓘ |
| status | open ⓘ |
| studiedBy | Paul Erdős NERFINISHED ⓘ |
| type | exponential Diophantine equation ⓘ |
| typicalNotation | ∑_{i=1}^{m-1} i^k = m^k ⓘ |
| unknowns |
k
ⓘ
m ⓘ |
| yearProposed | 1953 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.