Mersenne Primes
 

Completely new to this site and cannot even find my own post.
I have just posted a new question but cannot see it.
My theory on the existence of an infinite sequence of Mersenne Primes is not quite complete.
One more theorem needs proving: 'If phi(n) divides (n-1) then n is prime'.
Any comments?

[QUOTE=Stan;281096]Completely new to this site and cannot even find my own post.
I have just posted a new question but cannot see it.
My theory on the existence of an infinite sequence of Mersenne Primes is not quite complete.
One more theorem needs proving: 'If phi(n) divides (n-1) then n is prime'.
Any comments?[/QUOTE]

[url]http://www.mersenneforum.org/search.php[/url] as to the other theorem I looked at the thread but can't really help.

[QUOTE=Stan;281096]Completely new to this site and cannot even find my own post.[/QUOTE]

[url]http://mersenneforum.org/showthread.php?t=16293[/url]
?

I think the problem with direction comes from:

3p+2q =6x[SUB]n[/SUB]+1 where x[SUB]n[/SUB] = 4x[SUB]n-1[/SUB]+1 and x[SUB]1[/SUB]=0

s[SUB]0[/SUB]= 4 =3+1
s[SUB]n[/SUB]=s[SUB]n-1[/SUB][SUP]2[/SUP]-2

7=p^2-2=2*p+1 for p=3

so there are so many ways to connect it all together and work off different equations from 7 up you can use any of the 4 equations and then some to figure out the next step up algebraically. so with so many paths you can get so many results.

too many things I find with no useful looking properties:

[CODE](17:10)>S=X*P;print((S^2-2)%(2*P+1))
(1/4*X^2 - 2)[/CODE]

is my latest result problem is X is bigger than P all times except P=7 as far as I can tell. 14=2*7

194 = 1/4*2^2-2 = 1/4*4-2 = 1-2 = -1 mod 15

edit:[CODE](17:20)>S=X*P+y;print((S^2-2)%(2*P+1))
(1/4*X^2 - y*X + (y^2 - 2))[/CODE]

[QUOTE=science_man_88;281873]too many things I find with no useful looking properties:

[CODE](17:10)>S=X*P;print((S^2-2)%(2*P+1))
(1/4*X^2 - 2)[/CODE]

is my latest result problem is X is bigger than P all times except P=7 as far as I can tell. 14=2*7

194 = 1/4*2^2-2 = 1/4*4-2 = 1-2 = -1 mod 15

edit:[CODE](17:20)>S=X*P+y;print((S^2-2)%(2*P+1))
(1/4*X^2 - y*X + (y^2 - 2))[/CODE][/QUOTE]

z^2 = 0 or 1 mod 4 if I remember correctly , so doing the equation with modular arithmetic may work as 0 = 0 mod a.

[QUOTE=science_man_88;281873]too many things I find with no useful looking properties:

[CODE](17:10)>S=X*P;print((S^2-2)%(2*P+1))
(1/4*X^2 - 2)[/CODE]

is my latest result problem is X is bigger than P all times except P=7 as far as I can tell. 14=2*7

194 = 1/4*2^2-2 = 1/4*4-2 = 1-2 = -1 mod 15

edit:[CODE](17:20)>S=X*P+y;print((S^2-2)%(2*P+1))
(1/4*X^2 - y*X + (y^2 - 2))[/CODE][/QUOTE]

so y_n can be related back to y_1 the problem is if X_n could be and make the evaluation easier it would also allow direct computation of it an possibly render the system useless.

another possibility came up just playing around though it needs something still and it's a way to tell when 4m+3 ( technically a jump of 2 Mersenne numbers) divides 4n+2 or at least a ratio of them because these are what I get them down to:

31= 4(7)+3
14=4(3)+2

63=4(15)+3
194=4(48)+2

n=2n+1;m=m*[TEX]s_{x-2}[/TEX] with n originally set to 1, and m to 3.

if we can relate the m and n values I have a way using the squares of S already in use that I might be able to relate them back but I may just be complicating things.

[QUOTE=science_man_88;283231]another possibility came up just playing around though it needs something still and it's a way to tell when 4m+3 ( technically a jump of 2 Mersenne numbers) divides 4n+2 or at least a ratio of them because these are what I get them down to:

31= 4(7)+3
14=4(3)+2

63=4(15)+3
194=4(48)+2

n=2n+1;m=m*[TEX]s_{x-2}[/TEX] with n originally set to 1, and m to 3.

if we can relate the m and n values I have a way using the squares of S already in use that I might be able to relate them back but I may just be complicating things.[/QUOTE]

sorry [TEX]{s_{x-2}}^2[/TEX]

I'm told that changing the Horadam sequences equation to allow fractional coefficients couldn't come to make the Mersenne prime exponents , I'd just like more opinions on it.

I understand that when you write "Horadam sequence" you mean "linear recurrence relation of order 2". Such sequences can be linear or else exponential, in particular linear combinations of two algebraic numbers. (More is true: the numbers must be conjugates.)

If the Lenstra-Pomerance-Wagstaff conjecture is true, such a sequence can generate the Mersenne exponents only if it has 2^e^-gamma as a root. Using gp it's not hard to show that this does not happen for coefficients with fewer than 10,000 decimal digits. The basic command you want is
[code]algdep(2^exp(-Euler),2)[/code]

But that's moot, since basic linear algebra shows that no recurrence of order 2 can generate 2, 3, 5. You're sunk from the beginning.

You could try higher-order recurrences, but then you'll find that the highest order you can test for fails. (I'll let you do that as a basic exercise in matrix math, in gp or by hand.)