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).
All labels observed (2)
| Label | Occurrences |
|---|---|
| Erdős–Moser conjecture | 1 |
| Erdős–Moser equation canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896712 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Erdős–Moser equation Context triple: [Pál Erdős, knownFor, Erdős–Moser equation]
-
A.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
B.
Beal conjecture
The Beal conjecture is an unsolved problem in number theory proposing that if A^x + B^y = C^z with A, B, C, x, y, z positive integers and exponents greater than 2, then A, B, and C must share a common prime factor.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Erdős–Moser equation Target entity description: 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).
-
A.
Ramanujan–Nagell equation
The Ramanujan–Nagell equation is a famous Diophantine equation in number theory that has only finitely many integer solutions and is closely associated with the work of Srinivasa Ramanujan.
-
B.
Beal conjecture
The Beal conjecture is an unsolved problem in number theory proposing that if A^x + B^y = C^z with A, B, C, x, y, z positive integers and exponents greater than 2, then A, B, and C must share a common prime factor.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
- F. None of above. chosen
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
ⓘ
linked to:
Erdős–Moser equation
Prouhet–Tarry–Escott problem ⓘ 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
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| openQuestion | whether any nontrivial integer solutions exist ⓘ |
| proposedBy | Leo Moser ⓘ |
| 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
ⓘ
linked to:
Pál Erdős
|
| type | exponential Diophantine equation ⓘ |
| typicalNotation | ∑_{i=1}^{m-1} i^k = m^k ⓘ |
| unknowns |
k
ⓘ
m ⓘ |
| yearProposed | 1953 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Erdős–Moser equation Description of subject: 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).
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Erdős–Moser equation