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

MMMp=222p11,p1M_{M_{M_{p}}} = {2^2}^{2^{p - 1}} - 1, p \geq 1 and either a prime or a non-prime

p=1,222111=41=3p = 1, {2^2}^{2^{1 - 1}} - 1 = 4 - 1 = 3 is a prime

p=2,222211=161=15p = 2, {2^2}^{2^{2 - 1}} - 1 = 16 - 1 = 15 \neq a prime

p=3,222311=655361=65535p = 3, {2^2}^{2^{3 - 1}} - 1 = 65536 - 1 = 65535 \neq a prime

p=4,222411=1.15792089×10771=1.15792089×1077p = 4, {2^2}^{2^{4 - 1}} - 1 = 1.15792089×10^{77} - 1 = 1.15792089×10^{77} \neq a prime

p=5,222511=2655361p = 5, {2^2}^{2^{5 - 1}} - 1 = 2^{65536} - 1 \neq a prime:

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

Note by A Former Brilliant Member
3 months, 3 weeks ago

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@Yajat Shamji, why is it undefined? Its value is still 221612 ^ {2 ^ {16}} - 1

Aryan Sanghi - 3 months, 3 weeks ago

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Trying to find that out now, I have further simplified it to 26553612^{65536} - 1. @Aryan Sanghi

A Former Brilliant Member - 3 months, 3 weeks ago

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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?

Aryan Sanghi - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member 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 a2b2a ^ 2 - b ^2 which can be factored as (a + b)(a - b). I hope I explained well. :)

Aryan Sanghi - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member 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.

Aryan Sanghi - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member 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.

Aryan Sanghi - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@Aryan Sanghi You're right - I will edit it.

A Former Brilliant Member - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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

Aryan Sanghi - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava 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

Aryan Sanghi - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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You can read the long discussion we had in previous comment.

Aryan Sanghi - 3 months, 3 weeks ago

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

Aryan Sanghi - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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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).

Aryan Sanghi - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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I guess you should rather post a question on it.

Aryan Sanghi - 3 months, 3 weeks ago

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What should the question be? @Aryan Sanghi

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member Anything like whether a quadruple prime exists or how many quadruple primes are possible.

Aryan Sanghi - 3 months, 2 weeks ago

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@Aryan Sanghi Thanks - was waiting for a reply...

A Former Brilliant Member - 3 months, 2 weeks ago

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What should the question be? @Vinayak Srivastava

A Former Brilliant Member - 3 months, 2 weeks ago

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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.

Zakir Husain - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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I haven't discovered any prime as of yet.

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member Then????

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava I am working on it - currently 213107212^{131072} - 1 is undefined according to all calculators or is above their amount of numbers limit.

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member It is not prime. As @Aryan Sanghi said, 21310722^{131072} is a square and so it will be a difference of two squares. Also, ab1a-b\neq 1, so it will have at least 2 factors other than itself and 1.

Vinayak Srivastava - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@Vinayak Srivastava Not 2, but 20 + factors!

A Former Brilliant Member - 3 months, 3 weeks ago

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@Vinayak Srivastava There won't be any prime of such type, except when n=1n=1, then 22201=2211=41=32^{2^{{2^{0}}}}-1=2^{{2^1}}-1=4-1=3, which is prime. I don't know of any other case.

Vinayak Srivastava - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member Haha, I have corrected it, only one extra { messes things!

Vinayak Srivastava - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member :)

Vinayak Srivastava - 3 months, 3 weeks ago

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@A Former Brilliant Member It is impossible. I'll post a reason. Stay tuned.

Aryan Sanghi - 3 months, 3 weeks ago

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

Vinayak Srivastava - 3 months, 3 weeks ago

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@Aryan Sanghi I know 11 is not a prime - look at the note now!

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member The first sentence itself contradicts it!

p>1p>1

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava Good one! But look at my title...

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member I saw it, and instantly knew it isn't possible except 1, and 1 isn't prime!

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava Now, look at the first sentence..

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member Oh you changed it! :)

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava Yes - the first sentence actually contradicts the title!

A Former Brilliant Member - 3 months, 3 weeks ago

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@A Former Brilliant Member No problem, still a good effort!

Vinayak Srivastava - 3 months, 3 weeks ago

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

A Former Brilliant Member - 3 months, 3 weeks ago

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

Aryan Sanghi - 3 months, 3 weeks ago

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@A Former Brilliant Member I am no way criticizing you, don't feel that! I am glad and really appreciate your work!

Vinayak Srivastava - 3 months, 3 weeks ago

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@Vinayak Srivastava I wasn't talking about criticism.

A Former Brilliant Member - 3 months, 3 weeks ago

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