point is all it's saying is the next mersenne prime is 1.5 times previous:

davar55 look at [url]http://mathworld.wolfram.com/EberhartsConjecture.html[/url]

even I can find it.

as far as I can see davar all yours could add to his is k*1.5^n but that k would not be constant if i did the math correctly.

[QUOTE=science_man_88;250331]as far as I can see davar all yours could add to his is k*1.5^n but that k would not be constant if i did the math correctly.[/QUOTE]

I made one long multi-post in the Wagstaff thread.

Why don't you bring that post here?

[QUOTE=davar55;250337]I made one long multi-post in the Wagstaff thread.

Why don't you bring that post here?[/QUOTE]

all I gained from the PM's and that is what I said.

The YJ-Conjecture (in protoform)
 

Some excerpts from another thread:

Quote: Originally Posted by [B]davar55[/B]

[URL="http://www.mersenneforum.org/showthread.php?p=243337#post243337"][IMG]http://www.mersenneforum.org/images/buttons/viewpost.gif[/IMG][/URL]
T[I]wo points: First, neither the primes nor their logs are distributed
randomly. That's an approximation, there's no actual randomness
among the integers. Use randomness to make a conjecture at your
own risk. Second, if you check the subsequent analysis by jinydu,
the YJ-Conjecture is at least partially proven. It may eventually
be a lemma. Using a ratio of 3/2 = 1.500 is cleaner, leads to a
similar estimate of the next two gaps, and thus may help find
M47 and M48 and M49 and ...[/I]

Quote: Originally Posted by [B]davar55[/B]
[URL="http://www.mersenneforum.org/showthread.php?p=243392#post243392"][IMG]http://www.mersenneforum.org/images/buttons/viewpost.gif[/IMG][/URL]
Sin[I]ce you're rightfully looking for a proof, I can't give one to you yet.
My original data was based on computing Mn+1/Mn for
n = 1 to 39, seeing patterns in the ratio and difference gaps, eventually
intuiting that Mn ~ K*(3/2)^n. I then integrated this "Pythagorean"
looking value with what I knew in other simple number theoretic cases.
It seemed plausible. Then I wrote a program to compute this sequence,
for various values of K. Look how well M23 = 11213 fits my conjecture.

I think we'll see another such close match soon (i.e. in {M46, ... M60}),
which would give away the underlying math that connects this
conjecture/lemma to various other prime-related conjectures that you
all know very well.[/I]

Quote: Originally Posted by [B]davar55[/B]
[URL="http://www.mersenneforum.org/showthread.php?p=243402#post243402"][IMG]http://www.mersenneforum.org/images/buttons/viewpost.gif[/IMG][/URL]
[I]I'm in the process of coming back to speed after a several months long
layoff from my monograph and my mersenne-related math work. I'm also
getting my programming skills out of mothballs after a recent port to a
new Windows 7 computer (32-bit). I intend to convert my MPA calculator
from 8-bit byte arithmetic to 16-bit, and eventually overhaul my LL-tester
written in C based on a port-to-C of the F90 Fortran implementation.
This will take some time. It wouldn't be fair to me to put out a best
guess at this time just to demonstrate that my YJ-Conjecture has merit.[/I]

Y[I]es, since probabilities are non-negative.

And yes of course. I was referring to the sequence

P(n) = #Mprimes in [10^n,10^(n+1)-1] / (9 * 10^n)

which goes down to zero even though I say every P(n) > 0 so
that the sequence is unbroken and infinite.

This is close to the PNT, but using base ten, isn't it?[/I]


[I]Are you saying you can trivially prove my contention that
every number-of-digits has at least one representative
Mersenne Prime Exponent? And wouldn''t that immediately
imply proof of the mersennes' infinitude?[/I]


HERE IT IS !!!!!


[I]Getting back to the OP:

How about this three-parter conjecture:

Let Mn = the nth Mersenne Prime exponent (MPE).

(1) The ratios Rn = Mn+1/Mn are bounded above.

(2) In particular, 1 < Rn < 10 for all integral n >= 1.

(2') Consequently, there is at least one MPE for each number of
decimal digits > 0.

(2'') This implies the Mersenne Prime sequence is infinite.

(2''') This implies the Even Perfect Number sequence is infinite.

(3) The YJ-Conjecture:
lim (n->infinity) Rn= Mn+1/Mn = 3/2 = 1.500.

Take this as:

define function yj(K,M,N) = K * M^N

then there exists a real K in 0.5 < K < 2.0 and an M in 1 < M < 2 s.t.
the values of Rn hover around yj(K,M,N), i.e.
(similarly to the prime distribution function hovering around li(x))
the values of Rn grow like yj(K,M,N) and continue to
exceed it and then be exceeded by it infinitely often (cyclicly, i.e.
repeatedly), at varying intervals which may be estimated based on
the "best" values for K and M and for no other such values

(3') base M = 3/2 = 1.500, with best coefficient K t.b.d.
(possibly K = 1.0 or 2/3 or 4/3 or 3/2 or 2.0).

(Note especially MPE23 = 11213 for my best guess for K).

(3'') There's more, it's conjecturalisimo, and controversial.[/I]


Of course, there's more where that came from.

[QUOTE=davar55;250364]Some excerpts from another thread:
[/QUOTE]

once again this helps none for me except k*(3/2)^n = mn which is what I knew before you reposted this.

[QUOTE=science_man_88;250366]once again this helps none for me except k*(3/2)^n = mn which is what I knew before you reposted this.[/QUOTE]

You are not the only one reading this.

[QUOTE=davar55;250380]You are not the only one reading this.[/QUOTE]

yeah but I'm trying to understand it so i can try testing it.

[QUOTE=science_man_88;250400]yeah but I'm trying to understand it so i can try testing it.[/QUOTE]

May take you a while.

Try computing M46, M47, M48, M49, and M50,
based on the known M1 - M40, using different
values of K and M.

In an earlier post (#22) you computed a table based on Mn^(1/n).

The first column was 1.5^n as an approximation of MPEn.

To further test the conjecture, try computing the same table
using instead k*1.5^n for various values of k,
such as k = 1.00000, 4/3, 3/2, and 2.00000,
as I earlier suggested.

Then note the "close calls" such as (k=1,n=23), etc.

Then extrapolate (I sugested using splines, but I haven't
done that myself yet) to get a good fit for an MPE estimate.

If you want, for comparison to the Wagstaff stuff, it might be
interesting to do the same using their (or any other) base constant
and compare to using 1.500000 exactly.

BTW sure it's pythagorean. He was a bright guy in his day.

I only brought this up to ask a question.

Do the Wagstaff and Eberhart conjectures and their supporting evidence
depend on allowing for arbitrarily large ratios of consecutive Mersenne
Prime Exponents as the series grows (if it's infinite)?

Based on the admittedly small list generated so far (small in number,
not size of the MPs), it's hard to see the possibility of the ratio
exceeding, say, a million at some point, when thru M47* it never
exceeds 5. Of course, Small Numbers does leave this possible.
I wondered whether Wagstaff/Eberhart can be reconciled with
the first part of this conjecture (YJ) which says the ratio is bounded.