# Triple Mersenne Primes Proof where $p$ is either prime or non-prime and On the Matter of Triple Mersenne Numbers

$M_{M_{M_{p}}} = {2^2}^{2^{p - 1}} - 1, p \geq 1$ and either a prime or a non-prime

$p = 1, {2^2}^{2^{1 - 1}} - 1 = 4 - 1 = 3$ is a prime

$p = 2, {2^2}^{2^{2 - 1}} - 1 = 16 - 1 = 15 \neq$ a prime

$p = 3, {2^2}^{2^{3 - 1}} - 1 = 65536 - 1 = 65535 \neq$ a prime

$p = 4, {2^2}^{2^{4 - 1}} - 1 = 1.15792089×10^{77} - 1 = 1.15792089×10^{77} \neq$ a prime

$p = 5, {2^2}^{2^{5 - 1}} - 1 = 2^{65536} - 1 \neq$ a prime:

Therefore, there is no triple Mersenne prime where $p$ is a prime, however, when $p = 1$, a non-prime, it produces a prime. I have also discovered Triple Mersenne Numbers!

Note by A Former Brilliant Member
7 months, 2 weeks ago

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@Yajat Shamji, why is it undefined? Its value is still $2 ^ {2 ^ {16}} - 1$

- 7 months, 2 weeks ago

Trying to find that out now, I have further simplified it to $2^{65536} - 1$. @Aryan Sanghi

- 7 months, 2 weeks ago

It is very big, but that is defined, isn't it? Also, maybe at p = 11, it is a prime, but so big we can't evaluate, isn't it?

- 7 months, 2 weeks ago

What program you got? You sound very confident, @Aryan Sanghi.

- 7 months, 2 weeks ago

Actually above number is defined, codes can't evaluate that much big numbers accurately(I do code and that's why I am saying). Above number is composite for every positive integer p as it can be written of form $a ^ 2 - b ^2$ which can be factored as (a + b)(a - b). I hope I explained well. :)

- 7 months, 2 weeks ago

Also, I am using Alpertron's Integer Factorization Calculator - best IFC I have ever seen - uses Eliptical Curve Method as well as others.

- 7 months, 2 weeks ago

No calculator in the world can factor it as no code till now written can factor that much big numbers within 1day. 10^7 operations can be performed at max in one second and factoring it requires more than 10^50 operations.

- 7 months, 2 weeks ago

So, how long? Also, it's not a prime. I'll send the picture.

- 7 months, 2 weeks ago

Yes, I know it is not a prime. As I said for every positive integer p, it can be factored to a number of form (a + b)(a - b). So, actually for p = 4 also, it is not a prime.

- 7 months, 2 weeks ago

I will check but I believe that $p = 4$ is a prime - let me check first.

- 7 months, 2 weeks ago

You're right - I will edit it.

- 7 months, 2 weeks ago

You just checked for numbers 2,3,4,5 then how do you know that there can't be any prime after that?

- 7 months, 1 week ago

As it can be written of form a² - b² which can be factored into (a + b)(a - b)

- 7 months, 1 week ago

That is OK, but the proof in the note is not valid in my opinion.

- 7 months, 1 week ago

Yes, actually previously it was that at p= 4 it's prime. After that he changed it to that it is not prime after I told him. So, it's ok

- 7 months, 1 week ago

Just checking for some numbers is not a proof, I think it should be done algebraically.

- 7 months, 1 week ago

- 7 months, 1 week ago

@Yajat Shamji, good job!! I never thought of p = 1.

- 7 months, 1 week ago

@Aryan Sanghi, what was the proof you were telling of there? I wish to know it!

- 7 months, 1 week ago

Simply that 2^2^2^(p-1) can be written as a² and 1 can be written as 1² which can be factored to (a + 1)(a -1).

- 7 months, 1 week ago

I am making a note on Quadruple Mersenne Numbers and Primes. What do you think, @Aryan Sanghi, @Vinayak Srivastava?

- 7 months, 1 week ago

I guess you should rather post a question on it.

- 7 months, 1 week ago

What should the question be? @Aryan Sanghi

- 7 months, 1 week ago

Anything like whether a quadruple prime exists or how many quadruple primes are possible.

- 7 months, 1 week ago

Thanks - was waiting for a reply...

- 7 months, 1 week ago

What should the question be? @Vinayak Srivastava

- 7 months, 1 week ago

In mathematics it is not sufficient to check only some cases to prove something, either you check for all primes (which are infinite), or either you give a mathematical proof so that the statement can mathematically be accepted.

- 7 months, 1 week ago

@Zakir Husain, thank you so much for reminding me but I think you should ask @Aryan Sanghi. He has got the proof.

- 7 months, 1 week ago

- 7 months, 2 weeks ago

Will nobody comment on the fact that I have found Triple Mersenne numbers? @Vinayak Srivastava, @Aryan Sanghi

- 7 months, 1 week ago

Can you tell any such prime you have discovered? :)

- 7 months, 1 week ago

I haven't discovered any prime as of yet.

- 7 months, 1 week ago

Then????

- 7 months, 1 week ago

I am working on it - currently $2^{131072} - 1$ is undefined according to all calculators or is above their amount of numbers limit.

- 7 months, 1 week ago

It is not prime. As @Aryan Sanghi said, $2^{131072}$ is a square and so it will be a difference of two squares. Also, $a-b\neq 1$, so it will have at least 2 factors other than itself and 1.

- 7 months, 1 week ago

Can we stop talking about Triple Mersenne primes?! I have discovered Triple Mersenne numbers!

- 7 months, 1 week ago

Not 2, but 20 + factors!

- 7 months, 1 week ago

There won't be any prime of such type, except when $n=1$, then $2^{2^{{2^{0}}}}-1=2^{{2^1}}-1=4-1=3$, which is prime. I don't know of any other case.

- 7 months, 1 week ago

LaTeX, please! (I can't see the red bit.)

- 7 months, 1 week ago

Haha, I have corrected it, only one extra { messes things!

- 7 months, 1 week ago

Thank god! There is a Triple Mersenne Prime I am editing the note right now!

- 7 months, 1 week ago

:)

- 7 months, 1 week ago

It is impossible. I'll post a reason. Stay tuned.

- 7 months, 1 week ago

But what is the flaw in my comment? (I know it, you know it, I want @Yajat Shamji to see.)

- 7 months, 1 week ago

I know $1$ is not a prime - look at the note now!

- 7 months, 1 week ago

The first sentence itself contradicts it!

$p>1$

- 7 months, 1 week ago

Good one! But look at my title...

- 7 months, 1 week ago

I saw it, and instantly knew it isn't possible except 1, and 1 isn't prime!

- 7 months, 1 week ago

Now, look at the first sentence..

- 7 months, 1 week ago

Oh you changed it! :)

- 7 months, 1 week ago

Yes - the first sentence actually contradicts the title!

- 7 months, 1 week ago

No problem, still a good effort!

- 7 months, 1 week ago

Also, can we stop talking about Triple Mersenne primes?! I have discovered Triple Mersenne numbers!

- 7 months, 1 week ago

I really appreciate your efforts. As you can see in comments, I was at your side. Keep it up.:)

- 7 months, 1 week ago

I am no way criticizing you, don't feel that! I am glad and really appreciate your work!

- 7 months, 1 week ago