Mersenne Numbers: Definition

[R.D. Silverman]

In another thread, it was witten:


"also,one of the definitions of a mersenne number is that p is prime.another is that p is a positive integer.i could easily accuse you of not knowing your definitions. "


It is FALSE that another definition allows p to be any positive integer.
I have heard this false assertion many times. Repeating it does not
make it true.

If you will not accept me as an authority, perhaps you will accept:

P. Ribenboim: "The Book of Prime Number Records" 1st edition. p.75

"The numbers M_q = 2^q-1 (with q prime) are called Mersenne numbers

Or maybe the discussion in Hardy & Wright, sect. 2.4 will convince you.
Mersenne himself only concerned himself with 2^q-1 for q prime.

Or maybe definition 5 in D. Shanks' book will convince you?

etc. etc.

If anyone writes 2^n-1 for general n as a Mersenne number, they are
being sloppy and ignoring mathematical history.

Go to any number theory conference. Ask any number theorist there.

Mersenne was concerned with the question of when 2^q-1 is prime.
And this ONLY happens when q is prime.

[TimSorbet]

The following claim n must be an integer, or a positive integer: (and don't require n to be prime)
http://mathworld.wolfram.com/MersenneNumber.html
http://mersennewiki.org/index.php/Mersenne_number

The following claim n must be prime: (they're all from Silverman's post because I can't find any web sites that directly claim n must be prime)
Silverman
P. Ribenboim: "The Book of Prime Number Records" 1st edition. p.75
definition 5 in D. Shanks' book
the discussion in Hardy & Wright, sect. 2.4
"Mersenne was concerned with the question of when 2^q-1 is prime."

http://en.wikipedia.org/wiki/Mersenne_prime says that definitions vary
http://primes.utm.edu/mersenne/ only discusses Mersenne primes without particularly defining Mersenne numbers and whether n may be composite or not.
http://www.mersenne.org/various/math.php is unclear as it too is focused on Mersenne primes, but in my opinion it implies that n need not be prime. "Mersenne numbers are of the simple form 2^P-1, where P is the exponent to be tested. It is easy to prove that if 2^P-1 is prime, then P must be a prime. Thus, the first step in the search is to create a list of prime exponents to test." If Mersenne numbers required a prime n, why would it even bother to suggest that the first step is to find Mersenne numbers with a prime n?


Isn't this all really a bit pointless, though? It is purely a question of definition.
It simply does not matter whether n must be prime or not to be a "Mersenne number". It's almost always quite clear whether the author is speaking only of ns that are prime or of all ns. And we all know that in order to be a Mersenne prime, it must first be a Mersenne number.
Besides, it seems to me it should be understood by the fact that you're using p, since p and q are usually be used for primes, while n and k are usually used for any integer. (at least from my observations, this seems to be true)

[R.D. Silverman]

Originally Posted by Mini-Geek

The following claim n must be an integer, or a positive integer: (and don't require n to be prime)
http://mathworld.wolfram.com/MersenneNumber.html
http://mersennewiki.org/index.php/Mersenne_number

)

On-line sources are unreliable, notoriously error-prone, and should not
be trusted. The sources that I quoted are all from highly
respected NUMBER THEORISTS.

Trying to quote Wikipaedia as an authority on anything is a joke.
Although mathworld is somewhat more reliable, I doubt whether their
post on the subject was written by a number theorist.

And if you still don't believe me, then I suggest you ask Hugh Williams.
He is a world recognized authority on the history of computational (and
other) number theory.

Check to see what Ore's History of Theory of Numbers has to say...

This is mathematics. Having a precise definition DOES matter.

[Orgasmic Troll]

Originally Posted by R.D. Silverman

On-line sources are unreliable, notoriously error-prone, and should not
be trusted. The sources that I quoted are all from highly
respected NUMBER THEORISTS.

Trying to quote Wikipaedia as an authority on anything is a joke.
Although mathworld is somewhat more reliable, I doubt whether their
post on the subject was written by a number theorist.

And if you still don't believe me, then I suggest you ask Hugh Williams.
He is a world recognized authority on the history of computational (and
other) number theory.

Check to see what Ore's History of Theory of Numbers has to say...

This is mathematics. Having a precise definition DOES matter.

"Mersenne numbers are numbers of the form 2^n - 1 where n is a positive integer" is a precise definition.

You're arguing that having a consistent convention matters in mathematics, which is patently false.

If someone were to present a brilliant proof with regard to Mersenne primes and referred to Mersenne numbers as all numbers of the form 2^n - 1 with n a positive integer, nobody would care. If in 50 years the convention shifts to where leading number theorists use Mersenne number in the broader sense, the mathematics will not change.

[Brian-E]

Originally Posted by Orgasmic Troll

You're arguing that having a consistent convention matters in mathematics, which is patently false.

Are you sure? If mathematicians took this attitude to extremes and used their own pet conventions for all sorts of ideas, their contributions would be hard to read and understand no matter how carefully they defined everything first. It seems to me that we ought to respect established conventions. These may change over time but there is no need to hasten the process.

[axn]

I guess, many times you need to refer to numbers of the form 2^n-1 (n primes/composites). Instead of inventing a new term, people just conveniently re-use Mersenne numbers as a shortcut for it, the context making it clear whether it is all n's or prime n's.

[Orgasmic Troll]

Originally Posted by Brian-E

Are you sure? If mathematicians took this attitude to extremes and used their own pet conventions for all sorts of ideas, their contributions would be hard to read and understand no matter how carefully they defined everything first. It seems to me that we ought to respect established conventions. These may change over time but there is no need to hasten the process.

Simple solution: Don't take it to extremes.

Pretty good advice for most things in life too.

[Orgasmic Troll]

Originally Posted by axn

I guess, many times you need to refer to numbers of the form 2^n-1 (n primes/composites). Instead of inventing a new term, people just conveniently re-use Mersenne numbers as a shortcut for it, the context making it clear whether it is all n's or prime n's.

Also, are there any types of numbers referred to as [Blank] Numbers that are all prime?

I think "Mersenne Numbers" and "Mersenne Primes" together make for an aesthetically satisfying definition of numbers of the form 2^n - 1 and the subset of them that are prime.

[TimSorbet]

Originally Posted by Orgasmic Troll

Also, are there any types of numbers referred to as [Blank] Numbers that are all prime?

Besides "Prime Numbers"? (or "Odd Prime Numbers" or "Even Prime Number")
Um...not that I know of. They'd probably just be referred to as [Blank] Primes, even if there are no [Blank] Numbers that aren't [Blank] Primes.

[lfm]

Originally Posted by axn

I guess, many times you need to refer to numbers of the form 2^n-1 (n primes/composites). Instead of inventing a new term, people just conveniently re-use Mersenne numbers as a shortcut for it, the context making it clear whether it is all n's or prime n's.

It seems one common convention is to use the name of the variable to indicate its type. 2^n-1 implies n is any (positive) integer. 2^p-1 implies p is a prime.

[davieddy]

Bob deserves all the stick he is getting after
circumnavigating the temporary closure of the thread where
he was taking the piss out of someone (possibly stupider
than he is).

Lord Lucan