Coprime integers¶

In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8724
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Coprime integers is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1.

One says also is prime to or is coprime with . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3.

For Coprime integers, the abstraction is narrower than the article's general subject matter: a positive case must preserve In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

Numbers That Share Nothing

Two numbers are coprime if the only number that splits both of them into equal groups is 1. For example, 8 and 9: you can't split both into groups of 2, or 3, or any other same number. But 6 and 9 aren't coprime, because both split into groups of 3.

Numbers With Only 1 in Common

Two whole numbers are coprime, also called relatively prime, when the only positive number that divides evenly into both of them is 1. For example, 8 and 9 are coprime: 8 is divisible by 1, 2, 4 and 8, and 9 by 1, 3 and 9, and the only one they share is 1. Neither 8 nor 9 is a prime number, but they are still coprime to each other. On the other hand, 6 and 9 are not coprime, because 3 divides both.

Relatively Prime Pair

Two integers are coprime (also called relatively prime or mutually prime) if the only positive integer that divides both is 1. Equivalently, their greatest common divisor (GCD) is 1. This also means no prime number divides both of them. Being coprime is a relationship between two numbers, not a property of each one alone: 8 and 9 are coprime even though neither is prime, while 6 and 9 are not, since both are divisible by 3.

In number theory, two integers a and b are coprime, relatively prime, or mutually prime if the only positive integer dividing both is 1, equivalently if gcd(a, b) = 1. A direct consequence is that any prime dividing a does not divide b, and vice versa, so the two numbers share no prime factors. One also says a is prime to b, or coprime with b. Coprimality is a relation on pairs, not a property of either number individually: 8 and 9 are coprime although neither is prime, while 6 and 9 are not, because 3 divides both. The concept depends only on shared divisors, not on the size or primality of the numbers involved.

Structural Signature¶

Sig role-phrases:

Defining carrier — When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or .
Constitutive relation — A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.
Operating condition — The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
Recognition evidence — Every pair of congruence relations for an unknown integer , of the form and , has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo .
Admissible variation — Informally, the probability that any number is divisible by a prime (or in fact any integer) is for example, every 7^th integer is divisible by 7.
Characteristic consequence — Hence the probability that two numbers are both divisible by is and the probability that at least one of them is not is Any finite collection of divisibility events associated to distinct primes is mutually independent.
Failure boundary — Here refers to the Riemann zeta function, the identity relating the product over primes to is an example of an Euler product, and the evaluation of as is the Basel problem, solved by Leonhard Euler in 1735.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.
Not an over-broad reading. A set of integers can also be called coprime if its elements share no common positive factor except 1.
Not an over-broad reading. A stronger condition on a set of integers is pairwise coprime, which means that and are coprime for every pair of different integers in the set.
Not an over-broad reading. The set is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime.
Not automatically Euclid number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Coprime integers applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Notation and testing. In their 1989 textbook Concrete Mathematics, Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative notation a\perp b to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as in is prime to ).
Notation and testing. The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
Probability of coprimality. Here refers to the Riemann zeta function, the identity relating the product over primes to is an example of an Euler product, and the evaluation of as is the Basel problem, solved by Leonhard Euler in 1735.
Notation and testing. When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or .
Notation and testing. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.
Notation and testing. A set of integers can also be called coprime if its elements share no common positive factor except 1.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Coprime integers names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. The strongest recognition evidence in the frozen account is: Every pair of congruence relations for an unknown integer , of the form and , has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo . A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification A set of integers can also be called coprime if its elements share no common positive factor except 1. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Coprime integers compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.—and the practical consequence—hence the probability that two numbers are both divisible by is and the probability that at least one of them is not is Any finite collection of divisibility events associated to distinct primes is mutually independent. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.
Check operation and conditions. The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
Demand recognition evidence. Every pair of congruence relations for an unknown integer , of the form and , has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo .
Test variation. Change an implementation or setting while preserving informally, the probability that any number is divisible by a prime (or in fact any integer) is for example, every 7^th integer is divisible by 7.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Coprime integers transfers literally when a new case preserves the same carrier type, relation, and recognition test. In their 1989 textbook Concrete Mathematics, Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative notation a\perp b to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as in is prime to ). The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .

Beyond the home domain. No canonical parent is asserted for Coprime integers. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For example, in the case of two events, a number is divisible by primes and if and only if it is divisible by ; the latter event has probability If one makes the heuristic assumption that such reasoning can be extended to infinitely many divisibility events, one is led to guess that the probability that two numbers are coprime is given by a product over all primes,. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1; recognition evidence → Every pair of congruence relations for an unknown integer , of the form and , has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo

Applied / In Practice¶

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Notation and testing; invariant → In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1; boundary → the case exits the class when a set of integers can also be called coprime if its elements share no common positive factor except 1

Structural Tensions¶

T1 — Stable identity versus admissible variation. A set of integers can also be called coprime if its elements share no common positive factor except 1. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. A stronger condition on a set of integers is pairwise coprime, which means that and are coprime for every pair of different integers in the set. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The set is coprime, but it is not pairwise coprime since 2 and 4 are not relatively prime. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Coprime integers literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Coprime integers distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Coprime integers is structural-leaning. Its structural side is the repeatable organization summarized by In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, . Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. It further constrains recognition and variation through: The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, . Every pair of congruence relations for an unknown integer , of the form and , has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo .

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Coprime integers literal. Its documented scope includes the condition that In their 1989 textbook Concrete Mathematics, Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative notation a\perp b to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as in is prime to ). Another bounded application condition is that The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, . These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Informally, the probability that any number is divisible by a prime (or in fact any integer) is for example, every 7^th integer is divisible by 7.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Coprime integers. The reviewed identity is: In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Coprime integers sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Number-Theoretic Properties & Tests (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1?
Euclid number. An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Composite number. A positive integer greater than one that can be expressed as a product of two smaller positive integers. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Primefree Sequence. A nontrivial Fibonacci-type integer sequence begun from coprime composite seeds and proved to contain only composite terms, typically by a finite cover of periodic modular divisibility classes. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Coprime integers remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Coprime_integers (revision 1363371102).
Preserved source candidate: https://archive.org/details/atreatiseonarit05eatogoog
Preserved source candidate: https://www.staff.uni-mainz.de/pommeren/Cryptology/Classic/4_Cylinder/LongPeriods.html
Preserved source candidate: https://www.nsa.gov/Portals/70/documents/about/cryptologic-heritage/historical-figures-publications/publications/wwii/german_cipher.pdf
Preserved source candidate: https://www.ciphermachinesandcryptology.com/en/onetimepad.htm
Preserved source candidate: https://www.britannica.com/topic/Vernam-Vigenere-cipher

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.