Fermat primality test
E530310
The Fermat primality test is a probabilistic algorithm that checks whether a number is prime by verifying congruences derived from Fermat's little theorem.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fermat primality test canonical | 5 |
| Solovay–Strassen primality test | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570565 — 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: Fermat primality test Context triple: [Fermat's little theorem, usedIn, Fermat primality test]
-
A.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
-
B.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
-
C.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
-
D.
Trial Division
The Trial Division is a unit within the New York County District Attorney’s Office responsible for prosecuting criminal cases in court from arraignment through verdict.
-
E.
trial division
The trial division is the part of the Supreme Court of the Australian Capital Territory responsible for hearing and determining cases at first instance, including serious criminal and significant civil matters.
- 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: Fermat primality test Target entity description: The Fermat primality test is a probabilistic algorithm that checks whether a number is prime by verifying congruences derived from Fermat's little theorem.
-
A.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
-
B.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
-
C.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
-
D.
Trial Division
The Trial Division is a unit within the New York County District Attorney’s Office responsible for prosecuting criminal cases in court from arraignment through verdict.
-
E.
Trial Division
The Trial Division is the branch of the International Criminal Court responsible for conducting trials and determining the guilt or innocence of accused individuals in cases of serious international crimes.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Monte Carlo algorithm
ⓘ
primality test ⓘ probabilistic algorithm ⓘ |
| assumes | Fermat's little theorem holds for primes ⓘ |
| basedOn | Fermat's little theorem ⓘ |
| canBeRepeatedWith | multiple random bases ⓘ |
| canMisclassify | Carmichael numbers as probably prime ⓘ |
| canUseBase | small fixed bases for some ranges of n ⓘ |
| coreCondition | a^(n-1) ≡ 1 (mod n) ⓘ |
| defines |
Fermat liar base
ⓘ
Fermat witness base ⓘ |
| errorSide | primality side ⓘ |
| failsOn | Carmichael numbers for many bases ⓘ |
| falseNegativeMeaning | prime reported as composite does not occur ⓘ |
| falsePositiveMeaning | composite reported as probably prime ⓘ |
| guarantees | if a^(n-1) ≢ 1 (mod n) then n is composite ⓘ |
| hasAdvantage |
simple to implement
ⓘ
very fast for large integers ⓘ |
| hasDisadvantage |
can be fooled by Carmichael numbers
ⓘ
error probability not easily bounded without base restrictions ⓘ |
| hasErrorType | false positive ⓘ |
| hasInput |
integer base a with 1 < a < n
ⓘ
integer n > 2 ⓘ |
| hasProperty | one-sided error ⓘ |
| hasSpecialComposite | Carmichael number ⓘ |
| isDeterministicFor | prime inputs ⓘ |
| isExampleOf | randomized primality test ⓘ |
| isIntroducedInContextOf | computational number theory ⓘ |
| isLessReliableThan |
Miller–Rabin primality test
ⓘ
Solovay–Strassen primality test ⓘ |
| isPredecessorOf | strong probable prime tests ⓘ |
| isProbabilisticBecause | some composite numbers pass the test ⓘ |
| isRandomizedFor | composite detection with random bases ⓘ |
| isRelatedTo | Fermat pseudoprime ⓘ |
| isSimplerThan |
AKS primality test
ⓘ
Miller–Rabin primality test ⓘ
linked to:
Miller primality test
|
| isTaughtIn |
algorithms courses
ⓘ
computational number theory courses ⓘ |
| isUsedAs | fast preliminary primality filter ⓘ |
| isUsedIn | cryptographic key generation heuristics ⓘ |
| neverProduces | false negative for primality ⓘ |
| outputs | composite or probably prime ⓘ |
| repetitionEffect | reduces probability of error ⓘ |
| timeComplexity | polynomial in log n ⓘ |
| typicalBaseChoice | random integer in [2, n−2] ⓘ |
| usesConcept |
congruence
ⓘ
modular arithmetic ⓘ |
| usesOperation | modular exponentiation ⓘ |
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: Fermat primality test Description of subject: The Fermat primality test is a probabilistic algorithm that checks whether a number is prime by verifying congruences derived from Fermat's little theorem.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Fermat primality test