[QUOTE=3.14159;227056]Mersenne numbers do not have any patterns, they're randomly distributed and have no covering set of divisors (Besides having Proth numbers as factors)[/QUOTE]

mersenne numbers have the equation a(n) = 3a(n-1) - 2a(n-2) which is weird as it's the same equation to fit the sequences I just talked of starting with the respective prime exponent that doesn't give a prime.

(14:39) gp > findrec([1,3,7,15,31,63,127])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
3 d.f.

[CODE]41441994149199491949199941949494914919499419411441141114499941994199[/CODE]

68 digits, and only has square numbers for digits!

Update: [code]41449199414911994941149149491999144449499419114949194149949499414499419449919999191449199941949494914919499419411441141114499941994199[/code]

134 digits, and only composed of square numbers! HA!

[QUOTE=3.14159;227058][CODE]41441994149199491949199941949494914919499419411441141114499941994199[/CODE]

68 digits, and only has square numbers for digits!

Update: [code]41449199414911994941149149491999144449499419114949194149949499414499419449919999191449199941949494914919499419411441141114499941994199[/code]

134 digits, and only composed of square numbers! HA![/QUOTE]

if only there was a pattern you'd find a 266 digit one.

Like this?
[code]
114491994149119144419491191491149119149149194941149914914141141919149419199199991914491999419494949149194994194114411411144999419941994144919941491199494114914949199914444949941911494919414914949941449941944191494919144919194949491491949141941144114111449194191419111
[/code]

But it's not 266 digits!

[QUOTE=science_man_88;227057]mersenne numbers have the equation a(n) = 3a(n-1) - 2a(n-2) which is weird as it's the same equation to fit the sequences I just talked of starting with the respective prime exponent that doesn't give a prime.

(14:39) gp > findrec([1,3,7,15,31,63,127])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
3 d.f.[/QUOTE]

They do. But remember that this script only suggests what relations are possible, not that they apply to your sequence. If numbers of the form 2^n - 1 form a 3-term homogeneous linear recurrence relation (and they do!) then it is the one listed. But a sequence could fit a recurrence relation for some length, then deviate from it. For example, using the first six terms of A006261 generates a recurrence, but it's not the right one!

(It happens in that case that there still is a nice recurrence, but you need more terms to get it.)

[QUOTE=kar_bon;227064]Like this?
[code]
114491994149119144419491191491149119149149194941149914914141141919149419199199991914491999419494949149194994194114411411144999419941994144919941491199494114914949199914444949941911494919414914949941449941944191494919144919194949491491949141941144114111449194191419111
[/code]

But it's not 266 digits![/QUOTE]

very close though :lol: doubt I'm able to find the length of the next one with this though.

[QUOTE=CRGreathouse;227065]They do. But remember that this script only suggests what relations are possible, not that they apply to your sequence. If numbers of the form 2^n - 1 form a 3-term homogeneous linear recurrence relation (and they do!) then it is the one listed. But a sequence could fit a recurrence relation for some length, then deviate from it. For example, using the first six terms of A006261 generates a recurrence, but it's not the right one!

(It happens in that case that there still is a nice recurrence, but you need more terms to get it.)[/QUOTE]

okay well since these other seem to fit the same relation and are all of form 2x+1,2(2x+1)+1,etc they should all deviate the same way in my eyes so what would the odds for an exception to give you a sequence that works in the same way as the original being inserted into 2^p-1 with p prime will actually be prime ?. can you give those odds ?

[QUOTE=3.14159;227058][CODE]41441994149199491949199941949494914919499419411441141114499941994199[/CODE]

68 digits, and only has square numbers for digits!

Update: [code]41449199414911994941149149491999144449499419114949194149949499414499419449919999191449199941949494914919499419411441141114499941994199[/code]

134 digits, and only composed of square numbers! HA![/QUOTE]

Hmm, I wonder how rare such numbers are. Heuristics suggest about
[TEX]\frac43\cdot\frac{3^n}{n\log10}[/TEX]
n-digit examples; how well does this hold in practice?

There are 0, 3, 11, 12, 43, 94, 239, 566, 1710 such primes with 1, 2, ..., 9 digits. The formula predicts 2, 3, 5, 12, 28, 70, 181, 475, 1266 primes; not too bad, perhaps, though biased on the low side. Maybe with the next correction term, the -1 in the denominator? That gives 3, 3, 6, 13, 31, 76, 193, 502, 1331.

This is an asymptotic density of something like n^0.477, so maybe somewhat rarer than palindromatic primes.

[QUOTE=science_man_88;227069]okay well since these other seem to fit the same relation and are all of form 2x+1,2(2x+1)+1,etc they should all deviate the same way in my eyes so what would the odds for an exception to give you a sequence that works in the same way as the original being inserted into 2^p-1 with p prime will actually be prime ?. can you give those odds ?[/QUOTE]

I don't understand.

[QUOTE=science_man_88;227060]if only there was a pattern you'd find a 266 digit one.[/QUOTE]

Here's a 266-digit one:
[code]11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111494999[/code]

[CODE](14:37) gp > findrec([11,23,47,95,191,383,767])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
3 d.f.
%63 = [3, -2]~
(14:39) gp > findrec([29,59,119,239,479])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
1 d.f.
%66 = [3, -2]~
(14:39) gp > findrec([1,3,7,15,31,63,127])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).[/CODE]

what are the odds that 2^ (the primes in the first 2)-1 are prime (which would put them in the primes of the last sequence). answer this and not only can we predict Mersenne primes but double Mersenne primes.

[QUOTE=CRGreathouse;227072]Here's a 266-digit one:
[code]11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111494999[/code][/QUOTE]

is there a 530 digit one ? if so it would possibly be part of this idea i have lol.

[QUOTE=science_man_88;227075]is there a 530 digit one ? if so it would possibly be part of this idea i have lol.[/QUOTE]

If you look at post #881, we expect lots of 530-digit ones -- about 8 × 10[SUP]249[/SUP]. The first is
[code]11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111491141[/code]

[QUOTE=science_man_88;227073][CODE](14:37) gp > findrec([11,23,47,95,191,383,767])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
3 d.f.
%63 = [3, -2]~
(14:39) gp > findrec([29,59,119,239,479])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).
1 d.f.
%66 = [3, -2]~
(14:39) gp > findrec([1,3,7,15,31,63,127])
Recurrence relation is a(n) = 3a(n-1) - 2a(n-2).[/CODE]

what are the odds that 2^ (the primes in the first 2)-1 are prime (which would put them in the primes of the last sequence). answer this and not only can we predict Mersenne primes but double Mersenne primes.[/QUOTE]

I don't understand your question at all. If I wanted to test if a small number was the exponent of a Mersenne prime, I would check if it's in the listed members of A000043 (by a binary search). If I wanted to test if a large number was a Mersenne exponent, I'd check if it was a prime, trial-divide, look for some small factors with ECM, and then do a Lucas-Lehmer test on it. I don't have a special test for numbers of a certain form unless I can show an algebraic factor, but that would be caught by my tests anyway. It wouldn't speed up the search at all.

if we prove all exceptions that aren't knocked out by others have a sequence of the same form and that series of that form starting with an exception can't have a 2^p-1 prime then we can knock out all primes that have form of that have sequence that start at an exception.

[QUOTE=science_man_88;227079]if we prove all exceptions that aren't knocked out by others[/QUOTE]

What's an exception? What does "knocked out" mean? What are "others"?

I assume that "knocked out" means that, in some sense, the number is guaranteed to not be the exponent of a Mersenne prime. But what determines if it's knocked out or not?

[QUOTE=science_man_88;227079]have a sequence of the same form and that series of that form starting with an exception can't have a 2^p-1 prime[/QUOTE]

The same form as what?

[QUOTE=science_man_88;227079]then we can knock out all primes that have form of that have sequence that start at an exception.[/QUOTE]

Without definitions for the terms, this doesn't tell me much of anything.

[i]But[/i] let me try something. Let's say that a number is "knocked out" if it is a member of one of the sequences S1, S2, S3, ..., Sk. Further, let's suppose that the sequences are, like A055010, exponential integer sequences, where each term is roughly twice the one before it. And let's look at large numbers, say exponents at least a million.

Suppose we want to find all Mersenne exponents between 1,000,000 and 1,100,000 by this method. (We know from Slowinski & Gage that there are none, but we're testing this!) Now, by our assumptions, a given sequence Sj will remove about lg (1100000/1000000) ≈ 0.13 primes from the list. Since there are about 100,000 members, we'll need a minimum of > 720,000 sequences to clear the list.

Not one sequence or three sequences; not even a hundred. Three-quarters of a million sequences! And that's just to clear a small part of a small range; a larger one would take many more. Further, if there's overlap, you'll need more.

So if you don't want to use millions of sequences, make sure your method doesn't follow my assumptions.

I realized that my idea takes a lot but if we can come up with something it's like a sieve.

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=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:

[QUOTE=3.14159;227146]That code is pretty long. Are you sure you had no simpler way of writing it? :no:[/QUOTE]

On the contrary, I could have written it to be one-third the size if I had time.

[QUOTE=CRGreathouse]On the contrary, I could have written it to be one-third the size if I had time.
[/QUOTE]

Excellent.

It's rare that sm88 doesn't have much to contribute today.

[QUOTE=3.14159;227179]It's rare that sm88 doesn't have much to contribute today.[/QUOTE]

yeah well between guest research on my mom's cancer type and family history plus playing with CRG's substring code to check for his requirements it's hard the only thing i know about the code is glue isn't defined in Pari last I checked (and i did check)

[QUOTE=science_man_88]yeah well between guest research on my mom's cancer type and family history plus playing with CRG's substring code to check for his requirements it's hard the only thing i know about the code is glue isn't defined in Pari last I checked (and i did check)
[/QUOTE]

I've recently been messing with vk, to test the sieve's efficiency using factor-rich vs. prime numbers.

Prime numbers tend to have the most candidates eliminated. Using the primorials, however, I think the sieve met its match.

Ex: I defined the primorial function as p(n), where n is the nth prime.

Trying p(90), I don't think even 1/2 of the candidates would be eliminated if I were to sieve to 10[sup]9[/sup]

[QUOTE=3.14159;227183]I've recently been messing with vk, to test the sieve's efficiency using factor-rich vs. prime numbers.

Prime numbers tend to have the most candidates eliminated. Using the primorials, however, I think the sieve met its match.

Ex: I defined the primorial function as p(n), where n is the nth prime.

Trying p(90), I don't think even 1/2 of the candidates would be eliminated if I were to sieve to 10[sup]9[/sup][/QUOTE]

the definition only works if you have a high enough prime limit in one sense if you use the function prime(x) if it's greater than primelimit you'd have to adapt it a bit.

[QUOTE=science_man_88;227182]the only thing i know about the code is glue isn't defined in Pari last I checked (and i did check)[/QUOTE]

I defined it in post #901.

[QUOTE=3.14159;227183]I've recently been messing with vk, to test the sieve's efficiency using factor-rich vs. prime numbers.

Prime numbers tend to have the most candidates eliminated. Using the primorials, however, I think the sieve met its match.[/QUOTE]

Primorial bases will have more candidates than prime bases, but if you sieve to any reasonable level candidates from either will produce primes at the same rate at a given size.

[QUOTE=3.14159;227183]Ex: I defined the primorial function as p(n), where n is the nth prime.

Trying p(90), I don't think even 1/2 of the candidates would be eliminated if I were to sieve to 10[sup]9[/sup][/QUOTE]

I would have expected 77% of the candidates to be removed at that level, using Mertens' Theorem (and direct calculation with the first 24 primes).

[QUOTE=CRGreathouse]Primorial bases will have more candidates than prime bases, but if you sieve to any reasonable level candidates from either will produce primes at the same rate at a given size.
[/QUOTE]

It depends on one's definition of "reasonable". Some would say reasonable is 10[sup]6[/sup], others would say it's 10[sup]15[/sup].


[QUOTE=CRGreathouse]I would have expected 77% of the candidates to be removed at that level, using Mertens' Theorem (and direct calculation with the first 24 primes).
[/QUOTE]

Only way to find out is to try it for yourself. Give NewPGen a few seconds for a certain k-range for k * p(90) + 1. Or give vk a few minutes to come up with all the candidates.

I'm going to search for k * 1621![sup]2[/sup] + 1, where k is between 10500 and 80500. The amount of digits should be about.. 9008-9010. (9008 = 2[sup]4[/sup] * 563).

Well, I sieved to 700M. Apparently, there are still a load of candidates left.

[QUOTE=3.14159;227194]It depends on one's definition of "reasonable". Some would say reasonable is 10[sup]6[/sup], others would say it's 10[sup]15[/sup].[/QUOTE]

I grant that there are people who would consider either reasonable. My statement holds for both definitions, and indeed a wider range on both sides.

[QUOTE=3.14159;227194]Only way to find out is to try it for yourself. Give NewPGen a few seconds for a certain k-range for k * p(90) + 1. Or give vk a few minutes to come up with all the candidates.[/QUOTE]

If you're interested, go ahead. I'd rather spend the time looking up better estimates of the product (with or without the assumption of the RH).

[QUOTE=CRGreathouse]I grant that there are people who would consider either reasonable. My statement holds for both definitions, and indeed a wider range on both sides.
[/QUOTE]

Sieving efficiency doesn't improve much when one goes deeper than about 10[sup]12[/sup] or so.

[QUOTE=CRGreathouse]If you're interested, go ahead. I'd rather spend the time looking up better estimates of the product (with or without the assumption of the RH).
[/QUOTE]

It was merely a suggestion. I do not intend on doing so.

[QUOTE=3.14159;227196]Well, I sieved to 700M. Apparently, there are still a load of candidates left.[/QUOTE]

[code]ff(x)=1/log(x)/exp(Euler)
s=ff(700e6);forprime(p=2,90,s*=p/(p-1));s[/code]

[QUOTE=3.14159;227198]Sieving efficiency doesn't improve much when one goes deeper than about 10[sup]12[/sup] or so.[/QUOTE]

[code]> ff(1e12)/ff(1e9)
%1 = 0.7500000000000000000000000000[/code]
so going to 1e12 only removes about a quarter of the candidates left at 1e9.

[QUOTE=CRGreathouse]Well, I sieved to 700M. Apparently, there are still a load of candidates left.
[/QUOTE]

I was not making a reference to p(90); I was referring to 1621![sup]2[/sup]

[QUOTE=3.14159;227201]I was not making a reference to p(90); I was referring to 1621![sup]2[/sup][/QUOTE]

In that case,
[code]s=ff(700e6);forprime(p=2,1621,s*=p/(p-1));s
%1 = 0.3648162081214616551560298255[/code]
so about 5/8 of the candidates should be removed by sieving up to 700 million. To get to 2/3 you'd need 5 billion...

Well, I found a PRP somewhat soon: 12066 * 1621![sup]2[/sup] + 1

Record project: k * 15670! + 1 (≈59k digits)

k-range: 100k to 330k.

Odds: 1 in 7880 with no sieving. With sieving to 1.5*10[sup]9[/sup], about 1 in 3900 to 4500, assuming 1/2 of candidates are eliminated.

Sieving should take 5-10 minutes.

This is taking somewhat longer than I thought. I canceled and went for a smaller range after 33 minutes of nothing.

I'm going to settle between 50 and 100 million.

Settled for 315 million, testing: The odds are about 1 in 3890. So I expect a prime in 3890 * 4 minutes, or 1556 minutes, or within 26 hours.

Okay: I sieved for k * 48[sup]7890[/sup] + 1 and k * 1296[sup]5680[/sup] + 1 (Correction, still sieving for the latter. Up to 193 billion)

I have an idea: Make a variant of vk which sieves for primes in em2, where em2 is the set of integers such that n * m + 1 is divisible by a user-specified number.

Let's make a description: em2 gives the liftmod for Mod(-1, x)/(n), where n is any integer, and where x is any integer coprime to n.

em2 gives the set of integers liftmod(Mod(-1, x)/(n)) + a*x.

a will be a user-specified range, and so will the x and n for em2.

It's basically a sieve that searches for prime cofactors of k * n + 1.

Ex: 17228365081076784548444895 * 400[sup]200[/sup] + 1 = 45569 * 455603 * 755617 * 7757609 * p523

Well, vk can do that. But if you want a specialized program, write one! :smile:

I figure that all I have to do is make some substitutions/additions/deletions to vk to get it working.

[QUOTE=CRGreathouse]Well, vk can do that. But if you want a specialized program, write one!
[/QUOTE]

You'd have to depend on the odds, though.

Joining GIMPS.
 

I decided to join GIMPS: I was assigned the following number, since I chose trial factoring work: 2[sup]75325247[/sup] - 1. I'm trial factoring to 590295810358705651712, or 2[sup]69[/sup]

Is the 70M range the current interest for trial factoring, or was that a rare fluke?

Found a prime: 63483 * 10[sup]8490[/sup] + 1 (8495 digits?)

Also found
83529* 48[sup]7890[/sup] +1 (13270 digits)
468550* 1999[sup]5346[/sup] +1 (17652 digits)
515361* 1296[sup]5680[/sup] +1 (17686 digits)

Searches ongoing:

k * 2[sup]856780[/sup] + 1 (≈257920 digits)
k * 798336[sup]20160[/sup] + 1 (≈119000 digits)

[QUOTE=3.14159;227268]I figure that all I have to do is make some substitutions/additions/deletions to vk to get it working.[/QUOTE]

All you have to do, really, is remove the exponent.

[QUOTE=3.14159;227269]You'd have to depend on the odds, though.[/QUOTE]

I don't know what that means.

[QUOTE=CRGreathouse]All you have to do, really, is remove the exponent.
[/QUOTE]

No point in doing so. In fact, no changes at all should be made. The exponent should simply be set to 1.

[QUOTE=3.14159;227272]I decided to join GIMPS: I was assigned the following number, since I chose trial factoring work: 2[sup]75325247[/sup] - 1. I'm trial factoring to 590295810358705651712, or 2[sup]69[/sup]

Is the 70M range the current interest for trial factoring, or was that a rare fluke?[/QUOTE]

That seems to be the typical range for trial division. ECM isn't that far yet, as I recall, and the LL tests are just below the ECM range.

[QUOTE=CRGreathouse]That seems to be the typical range for trial division. ECM isn't that far yet, as I recall, and the LL tests are just below the ECM range.
[/QUOTE]

What range is ECM reserved for?

[QUOTE=3.14159;227278]No point in doing so. In fact, no changes at all should be made. The exponent should simply be set to 1.[/QUOTE]

That was the meaning of post #925: "vk can do that".

You could produce a specialized script without the exponent, removing the extra operation, but it won't save you much time.

Of course it will be slower to use vk like this: for a given size, it's much faster to use a power (say, n = 856780 or even 100) than a non-power (n = 1).

[QUOTE=3.14159;227281]What range is ECM reserved for?[/QUOTE]

Dunno, change your settings to ask for it and see. My understanding, rightly or wrongly, is that ECM is most needed right now -- that there's an increasingly small pool of ECM'd exponents ready for LL.

[QUOTE=CRGreathouse]That was the meaning of post #925: "vk can do that".

You could produce a specialized script without the exponent, removing the extra operation, but it won't save you much time.

Of course it will be slower to use vk like this: for a given size, it's much faster to use a power (say, n = 856780 or even 100) than a non-power (n = 1).[/QUOTE]

I only use vk when searching for larger bases than NewPGen would allow.

[QUOTE=CRGreathouse]Dunno, change your settings to ask for it and see. My understanding, rightly or wrongly, is that ECM is most needed right now -- that there's an increasingly small pool of ECM'd exponents ready for LL.
[/QUOTE]

I'm guessing the optimal settings are B1 = 1M?; B2 = 100M?

[QUOTE=CRGreathouse]I don't know what that means.
[/QUOTE]

Odds? Chances of a certain cofactor being prime? Based on the assumption that: The candidates have no factor smaller than what was sieved up to.

7.5% done with trial factoring for 2[sup]75325247[/sup] - 1. I should be finished in 9-11 hours.

Ongoing searches: k * 2[sup]93560[/sup] + 1 (≈28170 digits)

[QUOTE=3.14159;227288]I'm guessing the optimal settings are B1 = 1M?; B2 = 100M?[/QUOTE]

If you're ECMing on your own, basic guidelines are here:
[url]http://www.fermatsearch.org/ecm.html[/url]

You run a set number of curves at a given B1, then more at a higher B1, etc.

[QUOTE=3.14159;227289]Odds? Chances of a certain cofactor being prime? Based on the assumption that: The candidates have no factor smaller than what was sieved up to.[/QUOTE]

OK, but I still don't understand what you're telling or asking me here:

[QUOTE=3.14159;227269]You'd have to depend on the odds, though.[/QUOTE]

[QUOTE=CRGreathouse]You'd have to depend on the odds, though.
[/QUOTE]

I just answered that.

[QUOTE=CRGreathouse;227279]That seems to be the typical range for trial division. ECM isn't that far yet, as I recall, and the LL tests are just below the ECM range.[/QUOTE]

mersenne.org and PrimeNet are down right now so I can't verify, but IIRC ECM is only for very small Mersenne numbers, like p<1M, and is intended to find more factors when the primality is already known. ECM isn't cost-effective to run before an LL test, but it's good for finding factors when TF becomes too slow.
By the way, last I heard, TF has more people than needed, and DC needs more people. ECM is a side effort.

[QUOTE=Mini-Geek]mersenne.org and PrimeNet are down right now so I can't verify, but IIRC ECM is only for very small Mersenne numbers, like p<1M, and is intended to find more factors when the primality is already known. ECM isn't cost-effective to run before an LL test, but it's good for finding factors when TF becomes too slow.
By the way, last I heard, TF has more people than needed, and DC needs more people. ECM is a side effort.[/QUOTE]

That's strange. The GIMPS site works just fine for me.

[QUOTE=3.14159;227299]I just answered that.[/QUOTE]

I hope you got whatever information you needed.

[QUOTE=Mini-Geek;227300]mersenne.org and PrimeNet are down right now[/QUOTE]

[QUOTE=3.14159;227301]That's strange. The GIMPS site works just fine for me.[/QUOTE]

[url]http://downforeveryoneorjustme.com/mersenne.org[/url]

[QUOTE=3.14159;227301]That's strange. The GIMPS site works just fine for me.[/QUOTE]
[QUOTE=CRGreathouse;227304][url]http://downforeveryoneorjustme.com/mersenne.org[/url][/QUOTE]
It's working for me now. I didn't think to check downforeveryoneorjustme while it was down.

At most, I can have three searches that are running at their fastest. Or else there are severe slowdowns.

[QUOTE=CRGreathouse]I hope you got whatever information you needed.
[/QUOTE]

I did. There's no need for implicit insults anyway.

[QUOTE=3.14159;227306]At most, I can have three searches that are running at their fastest. Or else there are severe slowdowns.[/QUOTE]

This will depend mostly on your processor (and what settings you use). If you have a quad-core processor and enough non-sieve foreground tasks to fill one core (or N cores and enough other tasks to fill N-3 cores) then you'll get slowdown running more than 3 sieves just from lack of cycles. If you have large sieves that are in L3, and your processor has a shared L3, then once that gets filled you get a serious slowdown (probably worse than the above 25% slowdown; this one might be a 70% slowdown depending on how much spills out).

I don't have code for vk handy, but if you'll post your processor and the vk code I can tell you which of these is likely the problem (or if it's something else).

[QUOTE=3.14159;227307]I did. There's no need for implicit insults anyway.[/QUOTE]

I don't know why you interpret that as an insult. (You seem quick to take umbrage.) You posted something that made no sense to me (just as if you had posted "elephants have four legs": true, I suppose, but what are you telling me?). I asked for clarification, and you told me more things that seemed unrelated (that all members of order [i]Proboscidea[/i], which includes elephants, are quadrupeds). Rather than ask for further clarification, I simply stated that I hoped you found what you needed: if you didn't, then you'd surely clarify what you needed rather than what you said, which is what I wanted to know.

Apparently you found out what you wanted to know. I'm glad. I still don't know what that is, but now I don't have to care.

Anywho:

Sieved to 10[sup]12[/sup] for k * 756[sup]9570[/sup] + 1, and currently sieving for k * 4999[sup]5260[/sup] + 1 and k * 1296[sup]9256[/sup] + 1

Found the 28170-digit prime 570331 * 2[sup]93560[/sup] + 1

And, update on the placing: If I am to find a prime, it would now be [B]729th[/B] place. By the time I do find one, I might have dipped into the 1000s.

The minimum for the top 5000 is now 161k digits. Ouch to everyone who barely failed.

yeah i just downloaded prime95 to do something with all the downtime my computer (I did before I just find my PC too slow regardless of what I help with.)

right now I got 2 worker running with over 2900 Mb of ram and it's going at about .1% every 17 seconds with other stuff running.

[QUOTE=science_man_88]yeah i just downloaded prime95 to do something with all the downtime my computer (I did before I just find my PC too slow regardless of what I help with.)

right now I got 2 worker running with over 2900 Mb of ram and it's going at about .1% every 17 seconds with other stuff running[/QUOTE]

I can't run p95 with the sieving for the primes and the testing running at the same time. It slows the testing down to shit, and I can't run vk in PARI either, because it would slow things down severely as well. I got a 35% slowdown when using vk. (However, this is experienced whenever deep sieving using vk. It does no harm up until the 10[sup]7[/sup] range.)

Not to mention, the network is failing as well, which is why I don't post often anymore.

14-15% done both workers now. at this rate I'll never finish before my sister takes over my computer when she comes down to nag me about noise lol.

[QUOTE=science_man_88]14-15% done both workers now. at this rate I'll never finish before my sister takes over my computer when she comes down to nag me about noise lol.
[/QUOTE]

You'd never finish, because you would be assigned a random exponent every time you closed p95.

At the moment, my interest is personal records. Currently my personal record is 77285 digits, and I'm looking for either 119000 or 257920 digits.

[QUOTE=3.14159;227324]You'd never finish, because you would be assigned a random exponent every time you closed p95.

At the moment, my interest is personal records. Currently my personal record is 77285 digits, and I'm looking for either 119000 or 257920 digits.[/QUOTE]

you realise I could write them in wordpad and use them next time I open through the test menu right lol.

I have a variety of personal records for the many forms of primes:

[B]Proth[/B]: 22147 * 2[sup]256720[/sup] + 1. (77285 digits) + Largest prime discovered thus far by me.
[B]Fermat[/B]: None, all previously discovered.
[B]Twin[/B]: About 96 digits.
[B]Primorial[/B]: 466*7297#+1 (3124 digits), when k = 1; None: All previously discovered.
[B]Factorial[/B]: When k = 1; None, all previously discovered. When k > 1:
1364 * 4200! + 1 (13399 digits)
[B]Cofactor[/B]: p523 from 17228365081076784548444895 * 400[sup]200[/sup] + 1
[B]Generalized Fermat[/B]: None, all previously discovered.
[B]Generalized Proth[/B]: 207408 * 77906[sup]8192[/sup] + 1 (40078 digits)
[B]Primorial-based proths[/B]: Where b is a primorial number:
703 * p(125)[sup]66[/sup] + 1 (19104 digits)
[B]Factorial-based proths[/B]: Where b is a factorial number:
12066 * 1621![sup]2[/sup] + 1 (9008 digits) + Need to improve on that.
[B]Mersenne[/B]: None, all previously discovered.
[B]Cullen/Woodall[/B]: All previously discovered
[B]Generalized Cullen-Woodall[/B]: 4034 * 1500[sup]4034[/sup] + 1 (12817 digits)

[quote=3.14159;227324]You'd never finish, because you would be assigned a random exponent every time you closed p95.

At the moment, my interest is personal records. Currently my personal record is 77285 digits, and I'm looking for either 119000 or 257920 digits.[/quote]

[quote=science_man_88;227326]you realise I could write them in wordpad and use them next time I open through the test menu right lol.[/quote]
Prime95 saves its assignments in the worktodo.txt file; it also produces files labeled something like z******** (for LL tests), f******** (for factoring), etc. So as long as you shut down Prime95 correctly (Test>Exit), it will pick up exactly where you left off upon restart with no manual intervention.

Even if you don't shut it down "properly" (say, a power outage or something like that), you'll lose no more than 30 minutes of work; it updates the z*/f* files every 30 minutes automatically.

[QUOTE=mdettweiler;227331]Prime95 saves its assignments in the worktodo.txt file; it also produces files labeled something like z******** (for LL tests), f******** (for factoring), etc. So as long as you shut down Prime95 correctly (Test>Exit), it will pick up exactly where you left off upon restart with no manual intervention.

Even if you don't shut it down "properly" (say, a power outage or something like that), you'll lose no more than 30 minutes of work; it updates the z*/f* files every 30 minutes automatically.[/QUOTE]

I realized this just saying even if that didn't happen I know how to at least restart them lol. I've actually switched to stress testing my CPU etc. make sure it can handle it any thing I can do that may optimize it would be nice I'm on a Compaq Presario SR2050NX with new ram and 4GB of ram. ATI graphics card,Pentium D processor.

To help with organizing prime searches, I have the following idea: I will only seek out these types of primes:

1. [B]Proths[/B], where b is 2.
2. [B]Generalized Proths[/B], where b is any integer.
3. [B]Factorial-based proths[/B], where b is a factorial number.
4. [B]Primorial-based proths[/B], where b is a primorial number.
5. [B]Prime-based proths[/B], where b is a prime number.
6. [B]Primorial[/B], k * p(n) + 1
7. [B]Factorial[/B], k * n! + 1
8. [B]Generalized Cullen/Woodall[/B], k * b^k + 1
9. [B]Factorial Cullen/Woodall[/B], where b, optionally k, is a factorial number.
10. [B]Primorial Cullen/Woodall[/B], where b, optionally k, is a primorial number.
11. [B]Prime-based Cullen/Woodall[/B], where b is a prime number
12. [B]k-b-b[/B], numbers of the form k * b^b + 1
13. [B]Factorial k-b-b[/B], where b, optionally k, is a factorial number.
14. [B]Primorial k-b-b[/B], where b, optionally k, is a factorial number.
15. [B]Prime-based k-b-b[/B], where b is a prime number.
16. [B]Number, square, and fourth[/B], where n^1 + 1, n^2 + 1, and n^4 + 1 are all primes.
17. [B]Cofactor[/B], where this is the cofactor of any of the types of numbers listed here. Note: This is the most difficult find of all the forms of prime in this list.
18. [B]General arithmetic progressions[/B], k * n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length.
19. [B]Obsolete-tech-proven primes[/B], using the original PrimeForm or Proth.exe, or any other prime to prove primality of any type of prime listed here. Note: The prime must be at least 7500 digits in length. (Depends on whether or not you view Proth.exe and the original PrimeForm as obsolete.)

By far, the most difficult find is #17, Cofactor.

Finding a large prime cofactor of about 6500 digits would be more impressive than finding a Generalized Proth of 65000 digits.

Correction: [quote=3.14159]14. [b]Primorial k-b-b[/b], where b, optionally k, is a factorial number[/quote]

Primorial* number is what was meant here.

Also: Tomorrow: I'll only be looking for small primes, as I am not going to be on my main prime-finding machine.


So: Anyone else up for the challenge of posting a cofactor prime of at least 500 digits? (Note: The cofactor has to be of one of the 19 types of prime listed.)

[QUOTE=3.14159;227327][B]Cofactor[/B]: p523 from 17228365081076784548444895 * 400[sup]200[/sup] + 1[/QUOTE]

While recognizing the difficulty of factoring vs. simply proving primality, this would usually be expressed as a 23(?)-digit cofactor rather than a 523-digit cofactor.

[QUOTE=3.14159;227344]19. [B]Obsolete-tech-proven primes[/B], using the original PrimeForm or Proth.exe, or any other prime to prove primality of any type of prime listed here. Note: The prime must be at least 7500 digits in length. (Depends on whether or not you view Proth.exe and the original PrimeForm as obsolete.)[/QUOTE]

I don't think I'd want to encourage people to use outdated programs...

[QUOTE=CRGreathouse]While recognizing the difficulty of factoring vs. simply proving primality, this would usually be expressed as a 23(?)-digit cofactor rather than a 523-digit cofactor.
[/QUOTE]

Never said the divisors had to be small:

Ex: 499685426401622224432345842654646 * 400[sup]270[/sup] + 1 = 882597737519322866689219866761 * p706, new cofactor record.

[QUOTE=CRGreathouse]I don't think I'd want to encourage people to use outdated programs...
[/QUOTE]

The whining Luddites would love to disagree. And besides, it's kind of a challenge. Prove a decent-sixed number prime, with Proth.exe. It's only 1/4th the speed of PFGW.

[QUOTE=3.14159;227351]So: Anyone else up for the challenge of posting a cofactor prime of at least 500 digits? (Note: The cofactor has to be of one of the 19 types of prime listed.)[/QUOTE]

By your standards and with my usual method, I find a p501 in 203*2^1658+1. :razz:

[QUOTE=CRGreathouse]By your standards and with my usual method, I find a p501 in 203*2^1658+1.
[/QUOTE]

Let me guess: 3 * p501.

Okay, wiseguy, the smallest factor must be a p40, and the digits must follow a random distribution that is [B]statistically likely[/B]. Good luck.