[QUOTE=science_man_88;227086]I realized that my idea takes a lot but if we can come up with something it's like a sieve.[/QUOTE]

Here's what I would like to see, to start. Think of this as a test: once you pass, you should be able to convince others (like me!) to help you make the idea work. Until then, you have an [i]idea[/i] for an idea, rather than just an idea.

Find an infinite sequence a1 < a2 < a3 < ... for which:
1. There is no member a in the sequence such that 2^a - 1 is prime.
2. The sequence is easy to compute.*
3. There are many primes in the sequence.**
4. For each prime in the sequence, many (in the sense of #3) of them have no small prime factors.***

#1 means that the sequence filters out bad exponents, like you want. #2 means that the method is practical. #3 means that it's not trivial -- you're not saying something like "Mersenne exponents can't be composite". #4 shows that it's not trivial in another way: that you're just removing numbers with small prime factors.


* In particular, let's say: given the first n members of the sequence, the (n+1)-th term can be computed in polynomial time.
** Let's say: there are constants k and N such that, for every n > N, there are at least n/(log n)^k primes in the first n terms.
*** Say, 2^p - 1 has no prime factors below p[SUP]2[/SUP].

[QUOTE=3.14159;227094]Another try at predicting the next Mersenne? I think there are figures that place it at around 19M to 20M digits. I believe this was stated somewhere in the Prime Pages.[/QUOTE]

[url]http://www.nizkor.org/features/fallacies/gamblers-fallacy.html[/url]

[QUOTE=CRGreathouse]http://www.nizkor.org/features/falla...s-fallacy.html[/QUOTE]

O rly? Please explain how I made the Gambler's fallacy.

[QUOTE=3.14159;227097]O rly? Please explain how I made the Gambler's fallacy.[/QUOTE]

By expecting it to fall in a particular range based on it being 'due' in the Poisson model. This is exactly the classic mistake used to illustrate the gambler's fallacy! (Flipping coins and dropping the roulette ball are also examples of Poisson-distributed phenomena.)

[QUOTE=CRGreathouse]By expecting it to fall in a particular range based on it being 'due' in the Poisson model. This is exactly the classic mistake used to illustrate the gambler's fallacy! (Flipping coins and dropping the roulette ball are also examples of Poisson-distributed phenomena.)
[/QUOTE]

[URL=http://www.nizkor.org/features/fallacies/straw-man.html]Strawman[/URL], because I never made any expectations based on those figures. I merely mentioned that there are such figures.

It also seems that the sequence of prime numbers that are the concatenations of prime squares is not in the OEIS.. The smallest examples I could find are 499, 2549, and 12149.

A larger example is 3614928936149936120116936129268128929929841961.

[QUOTE=3.14159;227100][URL=http://www.nizkor.org/features/fallacies/straw-man.html]Strawman[/URL], because I never made any expectations based on those figures. I merely mentioned that there are such figures.[/QUOTE]

Any person claiming to "place [the next Mersenne exponent] at around 19M to 20M digits" is committing the gambler's fallacy. I didn't make any claims about what person made these claims. You brought it up, not me!

[QUOTE=3.14159;227107]It also seems that the sequence of prime numbers that are the concatenations of prime squares is not in the OEIS.. The smallest examples I could find are 499, 2549, and 12149.

A larger example is 3614928936149936120116936129268128929929841961.[/QUOTE]

I calculated the first 10,000 members of the sequence. I can't post them all, but here are the first few:
[code]449, 499, 1699, 2549, 4259, 4289, 4999, 8419, 9619, 9949, 12149, 25121, 25169, 25841, 25999, 28099, 28949, 34819, 41681, 41849, 42209, 42899, 43481, 43721, 44449, 45329, 46889, 49121, 49169, 49409, 49499, 49529, 49999, 52999, 53299, 68899, 84199, 91369, 92809, 92899, 93481, 94099, 94121, 94169, 94529, 94841, 94949, 94961, 94999, 96149, 96199, 98419, 99259, 99289, 99409, 99529, 99961, 102019, 118819, 121169, 121259, 121949, 136949, 136999, 169259, 169361, 184949, 184999, 187699, 251219, 252209, 252899, 253481, 254489, 255329, 256889, 257921, 259121, 259169, 259499, 259619, 259841, 259949, 280949, 289169, 289361, 289841, 289999, 327619, 348149, 361499, 361961, 372149, 388099, 411449, 417161, 422099, 429929, 436481, 437219, 439601, 444121, 444289, 444449, 444529, 444841, 448999, 462419, 468899, 469169, 473441, 491219, 493481, 493721, 495041, 496889, 497969, 499361, 504149, 529121, 529259, 529961, 529999, 532949, 532999, 624149, 624199, 688999, 841259, 841289, 910201, 916129, 916999, 922099, 925121, 925499, 925949, 925961, 928099, 932761, 940949, 942899, 944521, 944899, 949121, 949409, 949961, 961529, 961841, 979219, 984149, 984199, 992549, 992809, 993481, 994249, 994489, 994949, 995329, 998419, 999169, 999499, 999529, 999961, 1060949, 1135699, 1188149, 1213481, 1213721, 1214489, 1215299, 1215329, 1217921, 1219619, 1219949, 1219961, 1246099, 1276949, 1276999, 1346899, 1369169, 1369499, 1369961, 1513219, 1612999, 1681259, 1681289, 1691219, 1691369, 1691681, 1691849, 1694449, 1695041, 1695329, 1696241, 1699289, 1699361, 1699499, 1699619, 1716149, 1849259, 1876949, 1876999, 2125219, 2209169, 2209289, 2209499, 2209841, 2220199, 2294419, 2510609, 2511881, 2513699, 2517161, 2524649, 2529929, 2537219, 2542549, 2544121, 2544361, 2554289, 2558081, 2566049, 2576729, 2591681, 2592209, 2592899, 2594099, 2596241, 2596889, 2599999, 2656949, 2714419, 2809529, 2891219, 2893481, 2893619, 2893721, 2894449, 2895329, 2896241, 2899121, 2899499, 2899529, 2992949, 2992999, 3102499, 3276149, 3276199, 3481169, 3481259, 3481529, 3481999, 3588019, 3611369, 3611849, 3612019, 3612209, 3612809, 3612899, 3613481, 3613619, 3613721, 3615299, 3615329, 3616889, 3619289, 3619619, 3619961, 3721259, 3880949, 3880999, 3981619, 4100489, 4113569, 4124609, 4134499, 4136999, 4139129, 4143641, 4161299, 4171619, 4187489, 4196249, 4208849, 4212521, 4229441, 4254259, 4254361, 4254449, 4254949, 4254961, 4265699, 4271441, 4280999, 4292681, 4299209, 4299299, 4320419, 4332929, 4351649, 4368449, 4372499, 4388099, 4434281, 4441219, 4441849, 4442209, 4442549, 4443619, 4444169, 4444289, 4444949, 4446241, 4448419, 4449259, 4449409, 4449449, 4449619, 4489169, 4489999, 4491401, 4492549, 4494121, 4494169, 4494961, 4515299, 4528529, 4529449, 4552049, 4571219, 4591361, 4630019, 4691699, 4767299, 4789619, 4844561, 4858499, 4885481, 4896809, 4918769, 4922201, 4924649, 4926569, 4935089, 4944169, 4944361, 4944521, 4944899, 4944949, 4949449, 4950419, 4961449, 4963001, 4967219, 4972999, 4991681, 4991849, 4992809, 4993619, 4995041, 4995299, 4997921, 4998419, 4999121, 4999409, 4999529, 4999949, 4999961, 4999999, 5041121, 5041259, 5041529, 5041999, 5152949, 5285299, 5292809, 5293619, 5293721, 5294489, 5299409, 5329999, 5520499, 5712149, 5712199, 6241259, 6241289, 6241841, 6241999, 6300199, 6872419, 6889499, 6889529, 7276099, 7344149, 7672999, 7921169, 7921259, 7921289, 7921999, 8226499, 8411681, 8411849, 8412809, 8413619, 8413721, 8414449, 8417921, 8419121, 8419361, 9100489, 9106099, 9118819, 9121361, 9121529, 9121841, 9121999, 9146689, 9157609, 9169499, 9169529, 9169999, 9175561, 9177241, 9184949, 9196249, 9201601, 9218089, 9251219, 9251699, 9252209, 9252599, 9252809, 9254449, 9255041, 9258419, 9259121, 9259499, 9259619, 9259841, 9271441, 9289529, 9320419, 9326041, 9358801, 9368449, 9383161, 9409121, 9409529, 9410881, 9411449, 9416819, 9419321, 9422201, 9424949, 9424999, 9426409, 9427889, 9439601, 9442549, 9444949, 9452441, 9457121, 9462419, 9473441, 9479219, 9491401, 9491681, 9492209, 9493619, 9494999, 9496241, 9497921, 9499409, 9499619, 9499961, 9529259, 9529361, 9532949, 9564001, 9579121, 9612209, 9612899, 9613481, 9614489, 9615041, 9615329, 9616241, 9619289, 9619369, 9619409, 9619499, 9624149, 9662899, 9672149, 9672199, 9687241, 9769129, 9792149, 9796949, 9796999, 9829921, 9841841, 9912769, 9916129, 9928999, 9929929, 9934819, 9938809, 9941219, 9941699, 9943481, 9943721, 9944449, 9949169, 9949259, 9949619, 9952441, 9952949, 9953299, 9954289, 9961841, 9961961, 9968899, 9978961, 9984199, 9994099, 9994169, 9994289, 9994499, 9995329, 9996149, 9996241, 9997969, 9999289, ...[/code]

a(10000) = 4289449841.

What was the code for that one? Also: Found 436125841.

[QUOTE=3.14159;227117]What was the code for that one?[/QUOTE]

Ugh... wish you hadn't asked that one, the code is terrible. I'll post it, but I take no responsibility for your [url=http://www.youtube.com/watch?v=juFZh92MUOY]eyes[/url] or sanity.

[code]primesq(lim)={
my(u=[],v=vector(lim,n,prime(n)^2),mx=min(4444444449,nextprime(lim+1)^2),B,C,D,E,F,G,H,II);
print("Members up to "mx);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
if(isprime(B),
u=concat(u,B)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
if(isprime(C),
u=concat(u,C)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
if(isprime(D),
u=concat(u,D)
)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
for(e=1,#v,
E=glue(D,v[e]);
if(E>=mx,break);
if(isprime(E),
u=concat(u,E)
)
)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
for(e=1,#v,
E=glue(D,v[e]);
if(E>=mx,break);
for(f=1,#v,
F=glue(E,v[f]);
if(F>=mx,break);
if(isprime(F),
u=concat(u,F)
)
)
)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
for(e=1,#v,
E=glue(D,v[e]);
if(E>=mx,break);
for(f=1,#v,
F=glue(E,v[f]);
if(F>=mx,break);
for(g=1,#v,
G=glue(F,v[g]);
if(G>=mx,break);
if(isprime(G),
u=concat(u,G)
)
)
)
)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
for(e=1,#v,
E=glue(D,v[e]);
if(E>=mx,break);
for(f=1,#v,
F=glue(E,v[f]);
if(F>=mx,break);
for(g=1,#v,
G=glue(F,v[g]);
if(G>=mx,break);
for(h=1,#v,
H=glue(G,v[h]);
if(H>=mx,break);
if(isprime(H),
u=concat(u,H)
)
)
)
)
)
)
)
)
);
for(a=1,#v,
for(b=1,#v,
B=glue(v[a],v[b]);
if(B>=mx,break);
for(c=1,#v,
C=glue(B,v[c]);
if(C>=mx,break);
for(d=1,#v,
D=glue(C,v[d]);
if(D>=mx,break);
for(e=1,#v,
E=glue(D,v[e]);
if(E>=mx,break);
for(f=1,#v,
F=glue(E,v[f]);
if(F>=mx,break);
for(g=1,#v,
G=glue(F,v[g]);
if(G>=mx,break);
for(h=1,#v,
H=glue(G,v[h]);
if(H>=mx,break);
for(i=1,#v,
II=glue(H,v[i]);
if(II>=mx,break);
if(isprime(II),
u=concat(u,II)
)
)
)
)
)
)
)
)
)
);
vecsort(u,,8)
};

glue(a,b)={
a*10^digits(b)+b
};
addhelp(glue, "glue(a, b): Returns the (decimal) concatenation of a and b.");

digits(x)={
my(s=sizedigit(x)-1);
if(x<10^s,s,s+1)
};
addhelp(digits, "digits(n): Number of decimal digits in n. Sloane's A055642.");[/code]

You could easily improve the range of the code 10-fold with a minor modification (and indefinitely by repeating the large blocks of code, but that takes actual CPU time unlike the minor modification), but I didn't need it to get to the range I chose. Of course the code could have been written very differently, but this worked and I wasn't planning on keeping it.

[QUOTE=CRGreathouse]You could easily improve the range of the code 10-fold with a minor modification (and indefinitely by repeating the large blocks of code, but that takes actual CPU time unlike the minor modification), but I didn't need it to get to the range I chose. Of course the code could have been written very differently, but this worked and I wasn't planning on keeping it.
[/QUOTE]

That code is pretty long. Are you sure you had no simpler way of writing it? :no: