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.