theory on Mersenne primes ?

[science_man_88]

Originally Posted by ATH

Once I also thought I could find new insights with just some "handwaving" and "numbercrunching" and lucky guesses, but if it was so "easy" someone else would have already discovered it.

Mathematics have evolved so far now, and so many great minds have been working on it for centuries that most new discoveries only comes from deep study of theories by professional mathematicians. New proofs are now mostly hundreds of pages long and are so complicated it takes months or years for peers to review and accept them.

Of course there still *could* be simpler insights left to find just waiting for someone to think "outside the box" or view the problem from another angle, but the chance is small.

another thing I find weird is
that:

for(y=1,#mersenne,if(isprime(sum(X=1,#mersenne-(#mersenne-y),sumdigits(mersenne[X]^2))),print1(y",")))

is prime 20% of the time apparently and:

for(y=1,#mersenne,if(isprime(sum(X=1,#mersenne-(#mersenne-y),sumdigits(mersenne[X]))),print1(y",")))

is prime over 20% of the time. I would think this would seem like an unlikely amount but I once again may be wrong. I think it may come from the fact that 66% of 6x+1 and 6n-1 numbers have even sumdigits().

[CRGreathouse]

How often do you think this would happen by chance?

[science_man_88]

Originally Posted by CRGreathouse

How often do you think this would happen by chance?

Something tells me you know? I could give a guess with 1 or 2 double checks I know I can calculate it approximately in the odd placed Mersenne numbers which encompass most of the Mersenne primes. 6*(A002450)+1 = odd Mersenne numbers last I checked. using this fact and :

(09:04)>for(x=1,120,print1(sumdigits(6*x+1)","))
7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,7,4,1,

unless it hits the odd sumdigits an even amount they won't add up to an even gap between primes.

0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381,

the first few of A002450 :

if we plot the sumdigits based on modulo 3 we get:

(09:19)>c=vector(100);c[1]==0;for(i=2,#c,c[i]=c[i-1]*4+1)
(09:21)>c
%447 = [0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525,
(09:21)>c=c%3
%448 = [0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0,

which is 1,7,4, which only 33% that in my opinion should show up based on the even sumdigits alone though i realize that the odds could sum to an even every 2. Still I'm unsure why it would be 20% within a set of primes within the primes.

[CRGreathouse]

Originally Posted by science_man_88

Something tells me you know?

I'm sure I could come up with something. But really, until you have an idea of how often something would happen by chance you shouldn't say things like:

Originally Posted by science_man_88

is prime over 20% of the time. I would think this would seem like an unlikely amount but I once again may be wrong.

since you don't know what an "unlikely amount" would be.

It looks like you're examining the sum of the Mersenne exponents mod 9 (using 9 rather than 0 when 9 | n). Other than the 2nd member, these are {1, 2, 4, 5, 7, 8} and heuristically these would each occur with equal probability, for an average value of 4.5. The expected number of primes is then something like
[code]> sum(n=1,40,1/log(4.5*n))
%1 = 10.03235117441056250421202288[/url]
which is a bit higher (!) than observed but well within expected variance. So I would conclude that the number of primes you see is not unusual.

[science_man_88]

I've tried something weird instead of looking for a V_n(P,Q) to fit the exponents of mersenne primes I went another way and said lets replace the 0 and 1 of U_n(P,Q) with 2 and 3 and continue the best I've done so far is:

(10:43)>v=vector(30);v[1]=2;v[2]=3;for(i=3,#v,v[i]=(-1)*v[i-1]-(-4)*v[i-2]);v;
(10:43)>v
%841 = [2, 3, 5, 7, 13, 15, 37, 23, 125, -33, 533, -665, 2797, -5457, 16645, -

I was trying to go after the decimal place but i can't get that to work I was wondering if you knew anything I could do that might help this idea go further, I was thinking about combining V_n and U_n sequences if possible. anyone else with idea's, this idea came to me over time but I decided to try it harder with computer help after my argument with CRG in the math section.

[science_man_88]

I think I have an idea on how to extend it the problem is to figure out what it's helpfulness could be. Likely 0.

[science_man_88]

there's a method involved here anyone want to guess what it is ? No it's not just completely unrelated random guesses.

[science_man_88]

obviously sick of guessing:

this sequence I'm trying for is the Mersenne exponents in Lucas sequences as defined at http://en.wikipedia.org/wiki/Lucas_sequence to get it exact we have to stray from the definition. But in my guessing and thoughts on it I came across the fact that, if you have say 2 and 3 as the start numbers(I did) once you find one way to get a next value you like there are ways to keep it while changing the next value after it. for example to get the [2,3,5] I originally found P=3 Q=2 and starting with [2,3] you can get 5 on the end with those values, to keep it all you have to do it change P by 2 in this case and move Q by 3 you get P=1 Q=-1 if you move down then you get P=-1 Q=-4, which is the one I have now it still keeps the 5 then you alter it's path to change it to what you like. the hard part is I haven't found a way to stay in integer P and Q to get the Mersenne exponents. However if we break away that part and create a new set like Lucas sequences we can likely get it with decimal P and Q. all we should have to do then is find a way to predict a range of P and Q values on a line that I've found to go with my values, and we can then calculate limits to what the next in the sequence must be, if in fact it is a sequence.

[davar55]

There's a thread (over two years old) in Math or Puzzles here in which I
presented a conjecture of mine relating Fib and Lucas sequences.

I'll look for it later if you can't find it.

It needed a big numeric test that no one pursued here.
You might be interested.

[science_man_88]

Originally Posted by davar55

There's a thread (over two years old) in Math or Puzzles here in which I
presented a conjecture of mine relating Fib and Lucas sequences.

I'll look for it later if you can't find it.

It needed a big numeric test that no one pursued here.
You might be interested.

http://www.mersenneforum.org/showthr...ighlight=lucas I'm guessing.

[davar55]

Yes indeed you found it !

The relationship to primes might be mentioned
in an earlier thread in which I presented this
conjecture out of context.