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Nomenclature and notation

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This serves as a glossary for common terms and the notation that one may encounter in this Wiki.


Either Marin Mersenne or one of the special class of numbers that bear his name.

Mersenne number

Any number whether composite or prime of the form [math]\displaystyle{ 2^{x}-1 }[/math]. For one of these numbers to be prime, [math]\displaystyle{ x }[/math] (the exponent) must also be prime. Thus, the notation of [math]\displaystyle{ 2^{p}-1 }[/math] is generally used when discussing the search for a Mersenne prime.

Mersenne prime

A number that is itself prime and can be written in the form [math]\displaystyle{ 2^{x}-1 }[/math]. These are what GIMPS is searching for.

Mxx, M(x), and Mxx notation

Mxx can refer to one of 2 different things:

  • The xxth Mersenne prime, in order of size from the smallest to largest. This is the primary and most common usage or
  • The xxth Mersenne prime, in order of discovery (this usage is less common).

Generally Mxx would be the same in either case, but not so in the case of M45, M46, and M47. These were discovered in the order of M47, M45 (2 weeks later), then M46 (8 months later.). To avoid confusion, many speakers will use say "The xxth Mersenne prime found" or "The xxth known Mersenne prime" to indicate the second case listed above.

[math]\displaystyle{ M(x) }[/math] is normally used to denote [math]\displaystyle{ x }[/math] being run through the Mersenne function: [math]\displaystyle{ 2^{x}-1 }[/math].
[math]\displaystyle{ M_x }[/math] is sometimes used to denote the same function.

Confusion may sometimes occur when individuals refer to [math]\displaystyle{ 2^{61}-1 }[/math] or [math]\displaystyle{ 2^{127}-1 }[/math] or other known Mersenne primes as M61, M127, or such. The reader will generally understand that in cases where Mxx is larger than the largest known Mersenne prime, the writer is referring to Mexponent.

MMxx, MM(xx), and MMxx notation

The MMxx notation is used to refer to a 'Double Mersenne' of a number:

[math]\displaystyle{ 2^{(2^{xx}-1)}-1 }[/math]

Often the parenthesis are not used, but implied, such as MM61 referring to [math]\displaystyle{ 2^{(2^{61}-1)}-1 }[/math]

HPxx, HP(xx), or HPxx(yy)

HP[math]\displaystyle{ xx(yy) }[/math] refers to a Home prime. Unless noted with a subscript (for example [math]\displaystyle{ HPxx_8 }[/math]), it refers to a decimal or base 10 home prime. [math]\displaystyle{ xx }[/math] is the subject number (the one being tested or being discussed). [math]\displaystyle{ yy }[/math] is the step that is being tested or referred to.

HP[math]\displaystyle{ xx(yy)=zz }[/math] or HP[math]\displaystyle{ xx=zz }[/math] may be seen. [math]\displaystyle{ zz }[/math] can either be referring to the ulimate step value or the home prime for the subject number.