Skip to content

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

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.

Scope of Application

  • 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{Mp} can be prime only if.

  • Double Mersenne primes. Thus, the smallest candidate for the next double Mersenne prime is M{M{61}} , or 2 2305843009213693951 − 1.

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{Mp} = 2{2p-1}-1 where p is prime.

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 c5 is not prime, there is a chance to discover this by computing c5 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.

Abstract Reasoning

  1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, a double Mersenne number is a Mersenne number of the form M{Mp} = 2{2p-1}-1 where p is prime.
  3. Check operation and conditions. The first four terms of the sequence of double Mersenne numbers are.
  4. Demand recognition evidence. A double Mersenne number that is prime is called a double Mersenne prime.
  5. Test variation.

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.

Relationships to Other Abstractions

Local relationship map for Double Mersenne numberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.DoubleMersenne numberDOMAINPrime abstraction: Constraint — is a kind of, typicalConstraintPRIME

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

Neighborhood in Abstraction Space

Double Mersenne number sits in a sparse region of the domain-specific corpus (87th 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

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