Double Mersenne number¶

In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9057
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Double Mersenne number is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime.

In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime. The first four terms of the sequence of double Mersenne numbers are. A double Mersenne number that is prime is called a double Mersenne prime.

Since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M_{M_p} can be prime only if M p is itself a Mersenne prime. Thus, the smallest candidate for the next double Mersenne prime is M_{M_{61}} , or 2 2305843009213693951 − 1. this number is far too large for any currently known primality test.

For Double Mersenne number, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2^p-1}-1 where p is prime. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

The Twice-Done Number Trick

Here is a number trick: pick a number, multiply 2 by itself that many times, and take away 1. A double Mersenne number does the trick twice. Start with a special starting number like 2: the trick gives 3, then doing the trick again with 3 gives 7.

Mersenne of a Mersenne

A Mersenne number is what you get by taking a power of 2 and subtracting 1, like 2 times 2 times 2 minus 1, which is 7. A Double Mersenne Number does this trick twice: you start with a prime number p, make the Mersenne number 2^p - 1, and then use that answer as the new power: 2^(2^p - 1) - 1. If one of these is a prime number, it is called a double Mersenne prime. One can only be prime if the first-step number is itself prime. The next possible candidate is so gigantic that no known test can check whether it is prime.

Nested Mersenne Numbers

A Mersenne number has the form M_n = 2^n - 1. A double Mersenne number applies this twice: it is M_(M_p) = 2^(2^p - 1) - 1, where p is a prime. For example, p = 2 gives M_2 = 3 and then M_3 = 7. A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number M_n can only be prime if n is prime, M_(M_p) can only be prime if M_p is itself a Mersenne prime. That rules out many candidates, and the smallest remaining candidate for the next double Mersenne prime, M_(M_61), is far too large for any currently known primality test.

A double Mersenne number is a Mersenne number of the form M_(M_p) = 2^(2^p - 1) - 1 with p prime, that is, a Mersenne number whose exponent is itself a Mersenne number with prime exponent. Those that are prime are double Mersenne primes. Because M_n can be prime only when n is prime, M_(M_p) can be prime only when M_p is a Mersenne prime, which sharply restricts the candidates. The smallest candidate for the next double Mersenne prime is M_(M_61) = 2^2305843009213693951 - 1, far beyond the reach of any currently known primality test. What makes a number a double Mersenne number is this nested form with a prime starting exponent, not merely being large or being a Mersenne number.

Structural Signature¶

Sig role-phrases:

Defining carrier — Catalan discovered this sequence after the discovery of the primality of M_{127}=c_4 by Lucas in 1876. p.
Constitutive relation — However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation).
Operating condition — The first four terms of the sequence of double Mersenne numbers are.
Recognition evidence — A double Mersenne number that is prime is called a double Mersenne prime.
Admissible variation — Since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M_{M_p} can be prime only if M p is itself a Mersenne prime.
Characteristic consequence — Thus, the smallest candidate for the next double Mersenne prime is M_{M_{61}} , or 2 2305843009213693951 − 1.
Failure boundary — this number is far too large for any currently known primality test.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime.
Not an over-broad reading. However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation).
Not an over-broad reading. The first four terms of the sequence of double Mersenne numbers are.
Not an over-broad reading. A double Mersenne number that is prime is called a double Mersenne prime.
Not automatically Superperfect number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Double Mersenne number applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

The recursively defined sequence. Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge.
Examples. The first four terms of the sequence of double Mersenne numbers are.
Double Mersenne primes. A double Mersenne number that is prime is called a double Mersenne prime.
Double Mersenne primes. Since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M_{M_p} can be prime only if M p is itself a Mersenne prime.
Double Mersenne primes. Thus, the smallest candidate for the next double Mersenne prime is M_{M_{61}} , or 2 2305843009213693951 − 1.
Double Mersenne primes. this number is far too large for any currently known primality test.

Outside mathematics, logic, and 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 Double Mersenne number names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime. The strongest recognition evidence in the frozen account is: A double Mersenne number that is prime is called a double Mersenne prime. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation). so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Double Mersenne number compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—however, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation).—and the practical consequence—thus, the smallest candidate for the next double Mersenne prime is M_{M_{61}} , or 2 2305843009213693951 − 1. 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, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime.
Check operation and conditions. The first four terms of the sequence of double Mersenne numbers are.
Demand recognition evidence. A double Mersenne number that is prime is called a double Mersenne prime.
Test variation. Change an implementation or setting while preserving since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M_{M_p} can be prime only if M p is itself a Mersenne prime.
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 Double Mersenne number transfers literally when a new case preserves the same carrier type, relation, and recognition test. Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. The first four terms of the sequence of double Mersenne numbers are.

Beyond the home domain. No canonical parent is asserted for Double Mersenne number. 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¶

The first four terms of the sequence of double Mersenne numbers are. 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 mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime; recognition evidence → A double Mersenne number that is prime is called a double Mersenne prime

Applied / In Practice¶

A double Mersenne number that is prime is called a double Mersenne prime. 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 → Double Mersenne primes; invariant → In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime; boundary → the case exits the class when however, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation)

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation). 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. The first four terms of the sequence of double Mersenne numbers are. 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. A double Mersenne number that is prime is called a double Mersenne 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. Since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M_{M_p} can be prime only if M p is itself a Mersenne prime. 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. Catalan discovered this sequence after the discovery of the primality of M_{127}=c_4 by Lucas in 1876. p. 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 Double Mersenne number literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Double Mersenne number distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Double Mersenne number is structural-leaning. Its structural side is the repeatable organization summarized by In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime. Its framed side is the mathematics, logic, and 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 first four terms of the sequence of double Mersenne numbers are. 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 mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime. 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: Catalan discovered this sequence after the discovery of the primality of M{127}=c4 by Lucas in 1876. p. However, if c5 is not prime, there is a chance to discover this by computing c5 modulo some small prime p (using recursive modular exponentiation). It further constrains recognition and variation through: The first four terms of the sequence of double Mersenne numbers are. A double Mersenne number that is prime is called a double Mersenne prime.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Double Mersenne number literal. Its documented scope includes the condition that Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. Another bounded application condition is that The first four terms of the sequence of double Mersenne numbers are. 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—Since a Mersenne number M p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M{Mp} can be prime only if M p is itself a Mersenne prime.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry typically is a kind of Constraint.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Double Mersenne number. The reviewed identity is: In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime. 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.

Relationships to Other Abstractions¶

Current abstraction Double Mersenne number Domain-specific

Parents (1) — more general patterns this builds on

Double Mersenne number is a kind of, typical Constraint Prime

Double Mersenne number membership is the checkable existence condition n = 2⁽²p-1) - 1 for some prime p.

Hierarchy path (1) — routes to 1 parentless root

Double Mersenne number → Constraint

Neighborhood in Abstraction Space¶

Double Mersenne number sits in a sparse region of the domain-specific corpus (87^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Algebraic Structures & Order Relations (18 abstractions)

Nearest neighbors

Wall–Sun–Sun prime — 0.82
Octonion — 0.81
Integral part — 0.81
Coprime integers — 0.80
Binade — 0.80

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 mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime?
Superperfect number. Superperfect number denotes positive integer which equals half of the sum of the divisors of the sum of its divisors in number theory. Tell: Which entry's carrier, operation, and failure condition are satisfied?
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?
Scientific Notation. A positional-number representation that expresses a nonzero quantity as a significand multiplied by an integer power of a base—conventionally ten—with normalization separating scale from leading digits. 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 Double Mersenne number remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and 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/Double_Mersenne_number (revision 1330715968).
Preserved source candidate: http://primes.utm.edu/mersenne/index.html#unknown
Preserved source candidate: http://www.doublemersennes.org/mm61.php
Preserved source candidate: https://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071444-6/S0025-5718-1955-0071444-6.pdf
Preserved source candidate: https://archive.org/stream/nouvellecorresp01mansgoog#page/n353/mode/2up
Preserved source candidate: https://archive.org/details/historyoftheoryo01dick/
Preserved source candidate: http://www.hoegge.dk/mersenne/NMC.html#unknown
Preserved source candidate: http://anthony.d.forbes.googlepages.com/mm61.htm
Preserved source candidate: https://web.archive.org/web/20090208194031/http://anthony.d.forbes.googlepages.com/mm61.htm

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.