Twin Primes

Hi. All this is pretty new to me and my maths is weak so forgive me if this is a daft question.

What are the chances that M39 (or one of the others) is the lower half of a twin prime? ie. M39 + 2 is also prime. Is it even possible?

[jeff8765]

I am not sure but i do not think that it is possible. I say this because we know that M39 is not divisible by 3 because it is prime. M39+1 is not divisible by 3 either because it is a power of 2. So M39+2 must be divisible by 3.

[ET_]

Uhm... a Mersenne Prime should be a string of binary "1" . If you add 2 (that is "10") to a Mersenne Prime the result won't give you a string of ones. That is, you won't get a Mersenne Prime anymore.

Luigi

(where did I go wrong? :( )

[ET_]

where did I go wrong?

You referred to twin primes, not (obviously) to Mersenne twin primes that can't exist ops:
Next time I'll read the post and connect my brain before answering, sorry.

Luigi

Originally Posted by jeff8765

I am not sure but i do not think that it is possible. I say this because we know that M39 is not divisible by 3 because it is prime. M39+1 is not divisible by 3 either because it is a power of 2. So M39+2 must be divisible by 3.

Wow! That's neat! I've never noticed that powers of 2 are not divisible by 3. Can someone explain why?

(Maybe I should have posted this in the maths forum. Sorry.)

[asdf]

powers of 2 are only divisible by 2 because 2 is its only prime factor. For example, 8 is 2x2x2. 16 is 2x2x2x2, etc.

[asdf]

Also, I forgot to say that powers of any prime number are only divisible by that prime number, because it is its only prime factor. 81 = 3x3x3x3, 125 = 5x5x5, etc.

[Maybeso]

Maybe this has already been answered above and I got lost, but ...

What about the possibility of adjacent Mersennes (having Twin Primes as exponents) both being prime?

examples: M(5), M(7) and M(17), M(19).

So can there be more like that, or is there some relationship that prevents it?

I guess a formal way to ask it would be:
Can M(6k - 1) and M(6k + 1) both be prime for k > 3?

[Prime95]

Mp could be the upper number in a twin prime. Will Edgington kept data on this at:

http://www.garlic.com/~wedgingt/M3status.txt

It looks like no one has attacked M39 - 2

[guido72]

Originally Posted by ET_

Uhm... a Mersenne Prime should be a string of binary "1"

Ciao Luigi! Mi potresti spiegare perchè? Grazie e saluti! Guido

eng. version:
Hi Luigi! Could you explain why it happens? Thanks and regards! Guido

[ET_]

Ciao Luigi! Mi potresti spiegare perchè? Grazie e saluti! Guido

Each power of 2 minus one gives a binary string of "1"':

2^2 = 4 = 100b ---> 2^2 - 1 = 3 = 11b
2^3 = 8 = 1000b ---> 2^3 - 1 = 7 = 111b
2^5 = 32 = 100000b ---> 2^5 - 1 = 31 = 11111b
2^7 = 128 = 10000000 ---> 2^7 - 1 = 127 = 1111111b
...
2^n = 1(0)n b ---> 2^n - 1 = (1)n b

(1)n means a string of n "1"

Luigi