The KY-Conjectory
 

See the thread Wagstaff Conjecture in Puzzles or Math,
I'm not sure which.

-- davar55

The crux of the YJ or YJ-Eb Conjecture is contained in the meaning of

lim(n->infinity) (M sub n) to-the (1 over n) = 3/2 = 1.500000 exactly.

See the whole conjecture under Wagstaff Conjecture in Math.

This YJ-Conjecture was first presented here, in the Mersenne Forum,
by davar55 in 2006.

It was, I think, post #11 in a thread about our discovering M41.

Interesting that we're currently nearly proving the primality of
the first 41 Mersenne Primes.

If anyone in this forum is interested in math
and has any questions here, I'd be glad to try
to answer them.

Pardon me, but how is this relevant to Conjectures 'R Us? I have seen some of the chatter about this conjecture in other parts of the forum and it seems to relate to Mersenne primes, not the generalized Sierpinski/Riesel conjectures and related k*b^n+-c primes that are discussed here. Perhaps you posted in the wrong forum?

Edit: Ah, I think I see where you may have been confused. Conjectures 'R Us is not intended as a place for posting of conjectures of all kinds; rather, it's a distributed project attempting to prove the generalized Sierpinski/Riesel conjectures.

[QUOTE=mdettweiler;248516]Pardon me, but how is this relevant to Conjectures 'R Us? I have seen some of the chatter about this conjecture in other parts of the forum and it seems to relate to Mersenne primes, not the generalized Sierpinski/Riesel conjectures and related k*b^n+-c primes that are discussed here. Perhaps you posted in the wrong forum?

Edit: Ah, I think I see where you may have been confused. Conjectures 'R Us is not intended as a place for posting of conjectures of all kinds; rather, it's a distributed project attempting to prove the generalized Sierpinski/Riesel conjectures.[/QUOTE]

This YJ-Conjecture is in the Conjectures R Us sub-forum of
the Prime Search Projects sub-forum of MersenneForum Dot Org.

I suppose this thread might reasonably be moved to Math.

I have no objections.

[QUOTE=davar55;248624]This YJ-Conjecture is in the Conjectures R Us sub-forum of
the Prime Search Projects sub-forum of MersenneForum Dot Org.

I suppose this thread might reasonably be moved to Math.

I have no objections.[/QUOTE]
Okay, thanks--I've moved the thread to Math.

[QUOTE=mdettweiler;248626]Okay, thanks--I've moved the thread to Math.[/QUOTE]

Sure, no problem. That's where it belongs anyway.

I question that. I've taken the logical next step and moved it to Misc. Math.

[QUOTE=davar55;248386]The crux of the YJ or YJ-Eb Conjecture is contained in the meaning of

lim(n->infinity) (M sub n) to-the (1 over n) = 3/2 = 1.500000 exactly.

See the whole conjecture under Wagstaff Conjecture in Math.[/QUOTE]

how I interpret this is :

[CODE](09:53)>v=vector(10,n,if(n>1,2*v[n-1]+1,1))
%13 = [1, 3, 7, 15, 31, 63, 127, 255, 511, 1023]
(09:53)>for(n=1,#v,v[n]=v[n]^1/n)
(09:53)>v
%14 = [1, 3/2, 7/3, 15/4, 31/5, 21/2, 127/7, 255/8, 511/9, 1023/10]
(09:53)>for(n=1,#v,v[n]=v[n]-3/2)
(09:53)>v
%15 = [-1/2, 0, 5/6, 9/4, 47/10, 9, 233/14, 243/8, 995/18, 504/5][/CODE]

as you can see in the last v not all conform to v[n]<3/2, which blows the statement you supposedly lay your whole conjecture on to high heaven.even placing parentheses correctly I still get it false for both mersenne numbers and mersenne primes.

[QUOTE=davar55;248413]
It was, I think, post #11 in a thread about our discovering M41.[/QUOTE]

I've tried one part I've never seen anywhere but that post and this is what I got:

[CODE](10:48)>v=[1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]
%33 = [1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]
(10:49)>for(i=1,#v,v[i]=v[i]/(3/2)^i)
(10:49)>v
%34 = [2/3, 4/3, 56/27, 80/27, 992/243, 448/81, 16256/2187, 21760/2187, 261632/19683, 349184/19683, 4192256/177147][/CODE]

how this helps I have no idea but I have a pattern in the fractions namely they are of the form 2x/3^y. what else are you saying about these ?

[QUOTE=science_man_88;248936]how I interpret this is :

[CODE](09:53)>v=vector(10,n,if(n>1,2*v[n-1]+1,1))
%13 = [1, 3, 7, 15, 31, 63, 127, 255, 511, 1023]
(09:53)>for(n=1,#v,v[n]=v[n]^1/n)
(09:53)>v
%14 = [1, 3/2, 7/3, 15/4, 31/5, 21/2, 127/7, 255/8, 511/9, 1023/10]
(09:53)>for(n=1,#v,v[n]=v[n]-3/2)
(09:53)>v
%15 = [-1/2, 0, 5/6, 9/4, 47/10, 9, 233/14, 243/8, 995/18, 504/5][/CODE]

as you can see in the last v not all conform to v[n]<3/2, which blows the statement you supposedly lay your whole conjecture on to high heaven.even placing parentheses correctly I still get it false for both mersenne numbers and mersenne primes.[/QUOTE]

Try just making a list.

N MPE[sub]N[/sub] MPE[sub]N[/sub]^(1.500000),

and notice the fluctuating approach to and then around
the value 3/2.

[QUOTE=davar55;249211]
N MPE[sub]N[/sub] MPE[sub]N[/sub]^(1.500000),[/QUOTE]

[TEX]n\times MPE_N \times MPE_N^{3/2}[/TEX]

this is how I interpret that which means you are saying that a [TEX]\text {integer>=1}* \text {integer>1}* \text {integer>1}^{3/2} = 3/2 [/TEX] ?

? lets take the lowest case of 2 1*2*2^(1/5) which is around 5.65 if we say it has to be 1.5 you're way off are you saying that [TEX]\text {that answer} \div {MPE_{N+1}} = 3/2[/TEX] then I'm more likely to believe it using the first example which would bring it to about 1.89

Now be careful sm, don't assume that what davar writes is actually correct, even once you remove the obvious mistakes.

[QUOTE=CRGreathouse;249227]Now be careful sm, don't assume that what davar writes is actually correct, even once you remove the obvious mistakes.[/QUOTE]

if a mistake is obvious to me it's unlikely that the thing holds any actual facts.

[QUOTE=science_man_88;249230]if a mistake is obvious to me it's unlikely that the thing holds any actual facts.[/QUOTE]

Fair enough. You'll also note the lack of evidence -- the standard conjecture has better numerical evidence as well as a strong heuristic.

best i can do is get it to 7 through possible alteration. then it fails I've tried a number spiral for the last digits of all integers >=1 and then circled the digit where the exponents would land I have a few ideas but I don't have a concrete way to present them, maybe a spread sheet.

[QUOTE=davar55;249211]Try just making a list.

N MPE[sub]N[/sub] MPE[sub]N[/sub]^(1.500000),

and notice the fluctuating approach to and then around
the value 3/2.[/QUOTE]

Once again for me, having to hurry due to limited
computing time resources (though I'm no longer
riding a dinosaur), that post by me was wrong.
You really should think first before "blindly" following
anyone, including me.

The YJ-Conjecture limit formula is:

lim(n->infinity) M[sub]n[/sub] ^ (1/n) = 3/2 = 1.500000.

So the list I'm suggesting you compute is:

n, MPE[sub]n[/sub], MPE[sub]n[/sub]^(1.0 / n)

for n = 1 to 47

where MPE[sub]n[/sub] is the list

2,3,5,7,13,17,19,31,61,89,107,127,521,607, etc.

Then we can analyze the data I started with in 2006.

[QUOTE=davar55;249352]Once again for me, having to hurry due to limited
computing time resources (though I'm no longer
riding a dinosaur), that post by me was wrong.
You really should think first before "blindly" following
anyone, including me.

The YJ-Conjecture limit formula is:

lim(n->infinity) M[sub]n[/sub] ^ (1/n) = 3/2 = 1.500000.

So the list I'm suggesting you compute is:

n, MPE[sub]n[/sub], MPE[sub]n[/sub]^(1.0 / n)

for n = 1 to 47

where MPE[sub]n[/sub] is the list

2,3,5,7,13,17,19,31,61,89,107,127,521,607, etc.

Then we can analyze the data I started with in 2006.[/QUOTE]

[CODE](17:02)>for(i=1,#mersenne,print(mersenne[i]^(1.0/i)))
2.000000000000000000000000000
1.732050807568877293527446342
1.709975946676696989353108873
1.626576561697785743211232345
1.670277652334810394803652891
1.603521621512549282985464791
1.522926998218753393386968192
1.536102555746382080084096039
1.578955831901959581920696844
1.566531025721843309228146507
1.529288540910522460366821574
1.497328106146992604557789778
1.618033530221347008876089003
1.580517075546684774482807483
1.611108181063782432675943740
1.617850569292629664391427757
1.575929486505463156125518121
1.566240292511125441127408419
1.552329702833776473429342949
1.521548663732084805037571386
1.548184829102329494595765216
1.519502318204047920019162208
1.499943367067368910338914811
1.510610475500344547104696107
1.490936647375361341678740151
1.472006957225681423614603740
1.486484542508562273885684721
1.500637696861403754501242636
1.492483197974171054909171632
1.481464232274212955846795928
1.486226683055510708608763914
1.526577508382904969438978292
1.512950108495140761550758073
1.511472357027693313061657786
1.498263869087911248387534693
1.512948108348191840117822958
1.496720384272055787678110414
1.513872584170716147906235743
1.523357195557165263568423205
1.524240229323290374602268512[/CODE]

[CODE](17:06)>plot(i=1,#mersenne,mersenne[floor(i)]^(1/floor(i)),1.4,2)

2 ""'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''|
|: |
|: |
|: |
|: |
| : |
| : |
| : |
| : |
| __ |
| x |
| : |
| : "" |
| xx __ _ |
| x : "" |
| : xx__ : " xx__ |
| : _ : " x |
| xx " : xx xx _ "___ __ __xx
| "" "" xx _""x____ "" x |
| "" |
| |
1.4 |..............................................................|
1 40[/CODE]

with the equation he just posted it now becomes clear that he's pointing towards [TEX]\sqrt[n] {MPE_n}[/TEX] tends towards 1.5

[CODE]1.5 2 1.13 1.33
2.25 3 1.13 1.33
3.38 5 1.01 1.48
5.06 7 1.08 1.38
7.59 13 0.88 1.71
11.39 17 1.01 1.49
17.09 19 1.35 1.11
25.63 31 1.24 1.21
38.44 61 0.95 1.59
57.67 89 0.97 1.54
86.5 107 1.21 1.24
129.75 127 1.53 0.98
194.62 521 0.56 2.68
291.93 607 0.72 2.08
437.89 1279 0.51 2.92
656.84 2203 0.45 3.35
985.26 2281 0.65 2.32
1477.89 3217 0.69 2.18
2216.84 4253 0.78 1.92
3325.26 4423 1.13 1.33
4987.89 9689 0.77 1.94
7481.83 9941 1.13 1.33
11222.74 11213 1.5 1
16834.11 19937 1.27 1.18
25251.17 21701 1.75 0.86
37876.75 23209 2.45 0.61
56815.13 44497 1.92 0.78
85222.69 86243 1.48 1.01
127834.04 110503 1.74 0.86
191751.06 132049 2.18 0.69
287626.59 216091 2 0.75
431439.88 756839 0.86 1.75
647159.82 859433 1.13 1.33
970739.74 1257787 1.16 1.3
1456109.61 1398269 1.56 0.96
2184164.41 2976221 1.1 1.36
3276246.61 3021377 1.63 0.92
4914369.92 6972593 1.06 1.42
7371554.88 13466917 0.82 1.83
11057332.32 20996011 0.79 1.9
[/CODE]


this the first is the calculated value if my interpretation is correct the second is the actual value the third is (estimated / actual) ? The fourth is (actual/estimated) ? anyone see anything useful ?

[QUOTE=science_man_88;249373][CODE]1.5 2 1.13 1.33
2.25 3 1.13 1.33
3.38 5 1.01 1.48
5.06 7 1.08 1.38
7.59 13 0.88 1.71
11.39 17 1.01 1.49
17.09 19 1.35 1.11
25.63 31 1.24 1.21
38.44 61 0.95 1.59
57.67 89 0.97 1.54
86.5 107 1.21 1.24
129.75 127 1.53 0.98
194.62 521 0.56 2.68
291.93 607 0.72 2.08
437.89 1279 0.51 2.92
656.84 2203 0.45 3.35
985.26 2281 0.65 2.32
1477.89 3217 0.69 2.18
2216.84 4253 0.78 1.92
3325.26 4423 1.13 1.33
4987.89 9689 0.77 1.94
7481.83 9941 1.13 1.33
11222.74 11213 1.5 1
16834.11 19937 1.27 1.18
25251.17 21701 1.75 0.86
37876.75 23209 2.45 0.61
56815.13 44497 1.92 0.78
85222.69 86243 1.48 1.01
127834.04 110503 1.74 0.86
191751.06 132049 2.18 0.69
287626.59 216091 2 0.75
431439.88 756839 0.86 1.75
647159.82 859433 1.13 1.33
970739.74 1257787 1.16 1.3
1456109.61 1398269 1.56 0.96
2184164.41 2976221 1.1 1.36
3276246.61 3021377 1.63 0.92
4914369.92 6972593 1.06 1.42
7371554.88 13466917 0.82 1.83
11057332.32 20996011 0.79 1.9
[/CODE]


this the first is the calculated value if my interpretation is correct the second is the actual value the third is (estimated[COLOR="Red"](n+1)[/COLOR] / actual) ? The fourth is (actual/estimated) ? anyone see anything useful ?[/QUOTE]
correction

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.

Part of the conjecture says that for every number of digits (1,2,3,...)
there is at least one Mersenne PRIME Exponent (MPE). As supporting
evidence, I listed the smallest MPE's known for each number of digits.

2
13
107
1279
11213
110503
1257787
13466917

Every number of digits so far, except for 1 with the even 2, has
an MPE with a lead digit of 1, satisfying this part of the conjecture.
Of course, this is a short list if the sequence of Mersenne Primes
is infinite, and the 7th and 8th MPE's with lead digit 1 appear to be
growing a bit faster than the earlier ones, in which case they MIGHT
overflow into a number of digits without an MPE with a lead 1.

I venture to say this is a real pattern, although I haven't proven it.

This is the YJ-Conjecture:

[I]Let Mn = the nth Mersenne Prime exponent (MPE).[/I]

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

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

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

[I](2'') This would imply the Mersenne Prime sequence is infinite.[/I]

[I](2''') This would imply the Even Perfect Number sequence is infinite.[/I]

[I](3) [/I][I]lim (n->infinity) {Rn= Mn+1/Mn} = 3/2 = 1.500.[/I]

[I]Take this as: [/I]

[I]define function yj(K,M,N) = K * M^N[/I]

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

[I](3') base M = 3/2 = 1.500, with best coefficient K = 1.00. [/I]
[I](possibly K = 2/3 or 4/3 or 3/2 or 2.0).[/I]
[I](Note especially MPE[sub]23[/sub] = 11213. and (3/2)^23 = approximately 11223).[/I]

That is most probably false. The logarithmic gap between Mp's should vary the same as the (classical) gap between (classical) primes. Up to now we have like 5-6 of them for each decimal, in average, but there is no reason why this "base 10" should have anything to do with it. In fact, if [URL="http://mathworld.wolfram.com/WagstaffsConjecture.html"]Wagstaff conjecture[/URL] is true, then the n-th mersenne prime exponent is approximative 1.4756^n. This is the "average" gap, but it can be as small as 1 (i.e. two consecutive exponents result in mersenne primes), or it can be as big as 11, 12, etc, jumping out of the "next digit order".

[QUOTE=LaurV;324635]That is most probably false. The logarithmic gap between Mp's should vary the same as the (classical) gap between (classical) primes. Up to now we have like 5-6 of them for each decimal, in average, but there is no reason why this "base 10" should have anything to do with it. In fact, if [URL="http://mathworld.wolfram.com/WagstaffsConjecture.html"]Wagstaff conjecture[/URL] is true, then the n-th mersenne prime exponent is approximative 1.4756^n. This is the "average" gap, but it can be as small as 1 (i.e. two consecutive exponents result in mersenne primes), or it can be as big as 11, 12, etc, jumping out of the "next digit order".[/QUOTE]

The first statement (primes vs Mersene prime exponents) sounds
plausible, but AFAIK is unproved. So is Wagstaff. So is YJ.
On what grounds does one distiinguish between conflicting
non-self-contradictory conjectures?

That is why I said "most probably".

[QUOTE=davar55;349665]The first statement (primes vs Mersene prime exponents) sounds
plausible, but AFAIK is unproved. So is Wagstaff. So is YJ.
On what grounds does one distiinguish between conflicting
non-self-contradictory conjectures?[/QUOTE]

Heuristic strength?

[QUOTE=davar55;349665]The first statement (primes vs Mersene prime exponents) sounds plausible, but AFAIK is unproved. So is Wagstaff. So is YJ.
On what grounds does one distiinguish between conflicting
non-self-contradictory conjectures?[/QUOTE]
[url]http://correctpi.com/[/url] ?

Wow, this must rank high in the crank score metter Batalov

Heh, this one was frequently quoted on the [I]other[/I] forums where a certain Beal conjecture "prover" was making rounds before (and needless to say, again, recently). (Did you think he posts only here? Oh no. On dozens of them. Literally. And he posted all the same circumlocutory barrage everywhere he went.)

[QUOTE=Batalov;349725]Heh, this one was frequently quoted on the [I]other[/I] forums where a certain Beal conjecture "prover" was making rounds before (and needless to say, again, recently). (Did you think he posts only here? Oh no. On dozens of them. Literally. And he posted all the same circumlocutory barrage everywhere he went.)[/QUOTE]

Well, I haven't done anything like that with YJ, and since it's unproven,
I only support the conjecture as long as it remains unproven and
continues to provide anything like a good match to the succeeding
Mersenne prime exponents as they are discovered. If the conjecture
is proven wrong, or its predictions go way way far offtrack, I will
by necessity stop supporting it.

[QUOTE=CRGreathouse;349717]Heuristic strength?[/QUOTE]

I'm sure you're prepared to show that the YJ version is
heuristically not as strong as Wagstaff?

But really, I only claimed I had discovered the 3/2 pattern
independently (without knowledge of) the similar Eberhart
conjecture, and added the first part, that the ratio is bounded.

WHat I called the YJ-Eb conjecture (to give credit to the earlier
discover of the 3/2 heuristic) could be called by any other name,
I included my initial only because I had found the 3/2 relationship
independently (before I even looked into MPs much).

Both the YJ-Eb-KY Conjecture and the Wagstaff conjecture
are based on approximating the ratio of consecutive MPEs,
and are not in competition. Until we generate an actual algorithm
or formula for MPs, they are both just so helpful in predicting
the next MP in sequence.

The issue of infinitude of the MPs is addressed in JY-Eb-KY,
and I would think so also in Wagstaff.

[QUOTE=davar55;349782]I'm sure you're prepared to show that the YJ version is
heuristically not as strong as Wagstaff?[/QUOTE]

I would certainly say that your conjecture is not as sound as the Wagstaff conjecture! There's a strong heuristic there, and none at all that I can see for yours. There's [i]numerical[/i] evidence, which isn't strong enough to distinguish between the two, but no [i]heuristics[/i] you've provided.

[QUOTE=CRGreathouse;354845]I would certainly say that your conjecture is not as sound as the Wagstaff conjecture! There's a strong heuristic there, and none at all that I can see for yours. There's [i]numerical[/i] evidence, which isn't strong enough to distinguish between the two, but no [i]heuristics[/i] you've provided.[/QUOTE]

OK. Granted.

Note:

1.500000 ^ 48 ~= 283387333.

1.470000 ^ 48 ~= 107456345.

M48* = 57885161.

So using YJEbKY, it's under by a factor of ~5.

Using Wag, it's under by a factor of ~2.

ATM, Wag is considerably closer.

My overdue apologies to two forumites jiny** and **motl.

I seven years ago misapplied their interest in this conjecture
and attributed to them a belief in its veracity that I now believe
they did not have.

I believe it is necessary to make up for one's own mistakes.
I've made a few big ones on this forum, and this is the only way
for me to acknowledge that.

[QUOTE=davar55;363464]My overdue apologies to two forumites jiny** and **motl.

I seven years ago misapplied their interest in this conjecture
and attributed to them a belief in its veracity that I now believe
they did not have.

I believe it is necessary to make up for one's own mistakes.
I've made a few big ones on this forum, and this is the only way
for me to acknowledge that.[/QUOTE]

That was jinydu and _motl. Sorry about that.

By the way, has Wagstaff Conjecture been proven yet?

[QUOTE=davar55;401050]By the way, has Wagstaff Conjecture been proven yet?[/QUOTE]

No. It's unlikely to be proved in the near future.

[QUOTE=CRGreathouse;401075]No. It's unlikely to be proved in the near future.[/QUOTE]

And why is that? And exactly how unlikely?

Has anyone tried contesting this conjecture based on M40 thru M48?

Could someone please compute
R[sub]n[/sub] = (M[sub]n+1[/sub] / M[sub]n[/sub])
for n from 1 through 47 and compare this sequence of ratios
to 1.5000 versus say 1.47
and see if we might guess at decent bounds on a guess
for the next few values M49, M50, and M51 ?

I think you mean exponents' ratios and not mersenne's ratios, which are powers of two...
Different "numerologists" did this repeatedly, with no utility for the reality.
[URL="https://primes.utm.edu/notes/faq/NextMersenne.html"]This graphic[/URL] is "linear" we know that, but from here to predictions... well... :whistle:

[QUOTE=davar55;419800]Could someone please compute
R[sub]n[/sub] = (M[sub]n+1[/sub] / M[sub]n[/sub])
for n from 1 through 47 and compare this sequence of ratios
to 1.5000 versus say 1.47
and see if we might guess at decent bounds on a guess
for the next few values M49, M50, and M51 ?[/QUOTE]

[QUOTE=LaurV;419817]I think you mean exponents' ratios and not mersenne's ratios, which are powers of two...
Different "numerologists" did this repeatedly, with no utility for the reality.
[URL="https://primes.utm.edu/notes/faq/NextMersenne.html"]This graphic[/URL] is "linear" we know that, but from here to predictions... well... :whistle:[/QUOTE]

Of course Mn refers to the nth Mersenne exponent here.

I'm just asking someone to double check my conjecture's extension
beyond M1 thru M39 where I discovered it, to M40 thru M48.

[QUOTE=davar55;419836]Of course Mn refers to the nth Mersenne exponent here.[/QUOTE]
Ok. Then we have the mersenne numbers being [TEX]2^{M_1}-1,\ 2^{M_2}-1,\ 2^{M_3}-1[/TEX]etc. Clear. :razz:

Then what can't you check how much 57885161/43112609 is? Don't you have windows calculator?

[QUOTE=LaurV;419842]Ok. Then we have the mersenne numbers being [TEX]2^{M_1}-1,\ 2^{M_2}-1,\ 2^{M_3}-1[/TEX]etc. Clear. :razz:
Then what can't you check how much 57885161/43112609 is? Don't you have windows calculator?[/QUOTE]

I'm suggesting a list for 2 through 48 of the consecutive ratios of'
Mersenne exponents to see whether the KY conjecture of this thread
leads towards 1.5000 or the other value 1.47... from Wagstaff.

Need a loop over the M exponents.

Then make it. What's so difficult?
[CODE]
gp > \r mprimes.gph
gp > m=MERSENNE_EXPONENTS
%1 = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243
, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 371
56667, 42643801, 43112609, 57885161, 0]
gp > for(i=1,47,print(1.0*m[i+1]/m[i]))
1.500000000000000000000000000
1.666666666666666666666666667
1.400000000000000000000000000
1.857142857142857142857142857
1.307692307692307692307692308
1.117647058823529411764705882
1.631578947368421052631578947
1.967741935483870967741935484
1.459016393442622950819672131
1.202247191011235955056179775
1.186915887850467289719626168
4.102362204724409448818897638
1.165067178502879078694817658
2.107084019769357495881383855
1.722439405785770132916340891
1.035406264185201997276441217
1.410346339324857518632178869
1.322039166925707180603046317
1.039971784622619327533505761
2.190594619036852814831562288
1.026008876044999483950872123
1.127954934111256412835730812
1.778025506108980647462766432
1.088478707929979435220945980
1.069489885258743836689553477
1.917230384764531000904821406
1.938175607344315347102051824
1.281298192317057616270305996
1.194981131733980072939196221
1.636445561874758612333300517
3.502408707442697752335821483
1.135555910834404675234759308
1.463507917429281863740396285
1.111689817115298536238647720
2.128503885876036728269024058
1.015172260393297406341800559
2.307753385294188709320286744
1.931407297113139975329120745
1.559080745800987709362135372
1.144816651124825568056713249
1.080226378266827693437124570
1.170903692442939715156789628
1.071711309385290800674432333
1.140381737437803184681961327
1.147675624404094156238502232
1.010993579113644208216805064
1.342650383325212352608954842
gp >[/CODE]

[CODE]a=matrix(44,44,m,n,MeVec[m]/MeVec[n]);print(a)[/CODE]

now just find MeVec in the PARI command thread and add to it.

That alone doesn't show much, I agree.
But to check the KY-conjecture's value,
we need n, (Mn) ^ (1.0/n) for n from 1 to 48
to see whether, among other things,
this tends toward 1.5000 or thereabouts.

If it tends toward Wagstaff's 1.47... as n grows,
then only part of the KY may be true. If it tends
toward 1.500, the whole KY may be valid.

I know C and not pari and my compiler is not
currently available.

[QUOTE=davar55;419848]That alone doesn't show much, I agree.
But to check the KY-conjecture's value,
we need n, (Mn) ^ (1.0/n) for n from 1 to 48
to see whether, among other things,
this tends toward 1.5000 or thereabouts.

If it tends toward Wagstaff's 1.47... as n grows,
then only part of the KY may be true. If it tends
toward 1.500, the whole KY may be valid.

I know C and not pari and my compiler is not
currently available.[/QUOTE]

GP2C ?

[QUOTE=science_man_88;419850]GP2C ?[/QUOTE]

no comprendi gp2c.

[QUOTE=davar55;419882]no comprendi gp2c.[/QUOTE]

[url]http://pari.math.u-bordeaux.fr/pub/pari/manuals/gp2c/gp2c.html[/url]

It has the nice advantage of requiring both a PARI/GP installation and a C compiler.

Alright, if no one wants to help me confirm my conjecture, so be it.

I take that back, you did try to help, and it's my responsibility to pursue
this further. If M49* is confirmed and revealed, I'll re-tool and re-run my
old program.

This may help:
[YOUTUBE]ewS1Kkzl8mw[/YOUTUBE]
[YOUTUBE]vSdws014e4k[/YOUTUBE]
(Prof.Caffo kindly makes his course [URL="https://www.youtube.com/playlist?list=PLpl-gQkQivXjqHAJd2t-J_One_fYE55tC"]fully available[/URL] without registration.)

You can estimate the confidence interval for [TEX]\beta_1[/TEX]. And when you do, estimate how many data points you need to confidently establish that one slope fits data any better that the other. It will not matter if you have 48 data points or 49; you may rather need more than a hundred to get a decent probability of discerning these two close slopes.

Now, Wagstaff conjecture [I]has[/I] reasons whether you are capable to evaluate them or not. And "your" conjecture's reason is "it is 3/2 because it is 3/2"?

[QUOTE=Batalov;421907]This may help:...
(Prof.Caffo kindly makes his course [URL="https://www.youtube.com/playlist?list=PLpl-gQkQivXjqHAJd2t-J_One_fYE55tC"]fully available[/URL] without registration.)

You can estimate the confidence interval for [TEX]\beta_1[/TEX]. And when you do, estimate how many data points you need to confidently establish that one slope fits data any better that the other. It will not matter if you have 48 data points or 49; you may rather need more than a hundred to get a decent probability of discerning these two close slopes.

Now, Wagstaff conjecture [I]has[/I] reasons whether you are capable to evaluate them or not. And "your" conjecture's reason is "it is 3/2 because it is 3/2"?[/QUOTE]

Thanks for the video links - it'll take a little time for me to absorb them.

My conjecture has three basic parts, but the 3/2 comes from a close fit for
M1 thru M39 and a near exact match for M23. But if it's right, M50 will almost
have to be big, like over 100M, and like M51 around say 200M just to make up
for the ground it lost by the more recent finds in 30M - 55M.

The other two parts of the conjecture are:
the ratio of consecutive MP exponents is bounded (by ten),
and every number of digits in decimal is represented by
at least one MP exponent with lead digit equal to "1".

The 3/2 is very Pythagorean. I believe someday a
formula will be discovered with (3/2)^n the dominant term.