Carmichael number
E530314
A Carmichael number is a composite integer that nonetheless satisfies Fermat's primality test for all bases coprime to it, making it a classic example of a Fermat pseudoprime.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Alford–Granville–Pomerance proved in 1994 that there are infinitely many Carmichael numbers | 1 |
| Carmichael number canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570591 — 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: Carmichael number Context triple: [Fermat's little theorem, relatedConcept, Carmichael number]
-
A.
Blum integer
A Blum integer is a special type of composite number formed as the product of two distinct prime numbers each congruent to 3 modulo 4, widely used in cryptography and pseudorandom number generation.
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B.
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.
-
C.
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.
-
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.
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.
- 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: Carmichael number Target entity description: A Carmichael number is a composite integer that nonetheless satisfies Fermat's primality test for all bases coprime to it, making it a classic example of a Fermat pseudoprime.
-
A.
Blum integer
A Blum integer is a special type of composite number formed as the product of two distinct prime numbers each congruent to 3 modulo 4, widely used in cryptography and pseudorandom number generation.
-
B.
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.
-
C.
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.
-
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.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
number theory concept ⓘ |
| characterization |
n is Carmichael iff it is composite and for every integer a, a^n ≡ a (mod n)
ⓘ
n is Carmichael iff n is square-free, composite, and for every prime p dividing n, p−1 divides n−1 ⓘ |
| classification | subset of composite integers that are Fermat-liar to all coprime bases ⓘ |
| definition | a composite integer n such that a^(n−1) ≡ 1 (mod n) for all integers a coprime to n ⓘ |
| discoveredBy | Robert Daniel Carmichael ⓘ |
| discoveryYear | 1910 ⓘ |
| example |
1105
ⓘ
1729 ⓘ 2465 ⓘ 2821 ⓘ 561 ⓘ 6601 ⓘ |
| factorizationExample |
1105 = 5 × 13 × 17
ⓘ
1729 = 7 × 13 × 19 ⓘ 561 = 3 × 11 × 17 ⓘ |
| field | number theory ⓘ |
| growth | number of Carmichael numbers up to x grows faster than x^c for some c<1 ⓘ |
| isA |
Fermat pseudoprime
ⓘ
composite integer ⓘ |
| namedAfter | Robert Carmichael ⓘ |
| property |
every Carmichael number has at least three distinct prime factors
ⓘ
every Carmichael number is square-free ⓘ for each Carmichael number n and each prime p dividing n, n ≡ 1 (mod p−1) ⓘ for each Carmichael number n, λ(n) divides n−1, where λ is the Carmichael function ⓘ infinitely many exist ⓘ is a universal Fermat pseudoprime ⓘ is composite but behaves like a prime in Fermat’s little theorem for coprime bases ⓘ no product of two distinct primes is a Carmichael number ⓘ passes Fermat primality test to every base coprime to it ⓘ set has asymptotic density 0 among positive integers ⓘ violates the converse of Fermat’s little theorem ⓘ |
| relatedConcept | Carmichael function ⓘ |
| relatedTo |
Fermat primality test
ⓘ
Fermat’s little theorem ⓘ
linked to:
Fermat's little theorem
composite number ⓘ prime number ⓘ pseudoprime ⓘ strong pseudoprime ⓘ |
| relation |
generalization of specific Fermat pseudoprimes to all coprime bases
ⓘ
subset of Fermat pseudoprimes ⓘ |
| smallestElement | 561 ⓘ |
| symbolicDefinition | n is Carmichael iff n is composite and a^(n−1) ≡ 1 (mod n) for all a with gcd(a,n)=1 ⓘ |
| theorem |
Alford–Granville–Pomerance proved in 1994 that there are infinitely many Carmichael numbers
ⓘ
linked to:
Carmichael number
|
| use |
demonstrates limitations of Fermat primality test
ⓘ
used as counterexamples in primality testing ⓘ |
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: Carmichael number Description of subject: A Carmichael number is a composite integer that nonetheless satisfies Fermat's primality test for all bases coprime to it, making it a classic example of a Fermat pseudoprime.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Carmichael number
→
theorem
→
Alford–Granville–Pomerance proved in 1994 that there are infinitely many Carmichael numbers
ⓘ
linked to: Carmichael number