I just realised it's only when i try files i think.

got it working on C lol nice.

[QUOTE=science_man_88;229943]well I can't go above 127 without the error and that's when it comes up lol.[/QUOTE]

...because below that point you're not using MPQS.

it let me make folders but not files even though I used to all the time I think lol so i was able to make the temp folder in C yeah lol.

I'd still like to get it to work for all portables as well then I can have a script to change it on the fly so I can make them bigger and bigger lol. actually sounds like too much time to do it lol.

[QUOTE=science_man_88;229978]I'd still like to get it to work for all portables as well then I can have a script to change it on the fly so I can make them bigger and bigger lol. actually sounds like too much time to do it lol.[/QUOTE]

Well, I did give you a file... did you try it?

[QUOTE=CRGreathouse;229980]Well, I did give you a file... did you try it?[/QUOTE]

I think it couldn't create it in the other drives on it's own like write can so I'll have to set it up first I guess then i think the main thing is changing the TEMP variable on the fly.

[QUOTE=science_man_88;229981]I think it couldn't create it in the other drives on it's own like write can so I'll have to set it up first I guess then i think the main thing is changing the TEMP variable on the fly.[/QUOTE]

That's what my batch file does: changes the TEMP variable (temporarily!), then runs gp.

[QUOTE=CRGreathouse;229982]That's what my batch file does: changes the TEMP variable (temporarily!), then runs gp.[/QUOTE]

is there a code in Pari that could read and change the TEMP setting of that file on the fly ? if so I can use all my drives possibly if we can write part in one place then the other so breaking it up lol I have C:,E:F:,and K: drives not to mention I might be able to find some blank CD's lol.

[QUOTE=science_man_88;229983]is there a code in Pari that could read and change the TEMP setting of that file on the fly ?[/QUOTE]

No. It tries not to mess with the user's operating system...

[QUOTE=science_man_88;229983]if so I can use all my drives possibly if we can write part in one place then the other so breaking it up[/QUOTE]

What you really want is to have it use only one drive, in particular your fastest-performing drive (that allows you to read and write).

[QUOTE=CRGreathouse;229984]No. It tries not to mess with the user's operating system...



What you really want is to have it use only one drive, in particular your fastest-performing drive (that allows you to read and write).[/QUOTE]

once I created tmp I could in that folder so that helps I don't quite know which ones fastest I k is small and an ipod f is a portable stick E is a portable that has the same size as my main drive and C is partially filled up because I've used it lol

I gave up over 55 minutes into Aliquot(276) should of made it print on interrupt as well lol I was going to Aliquot(276) with a<2000

[QUOTE=science_man_88;229988]I gave up over 55 minutes into Aliquot(276) should of made it print on interrupt as well lol I was going to Aliquot(276) with a<2000[/QUOTE]

For large factorizations you want to use some other program. msieve is good.

[QUOTE=Charles]For large factorizations you want to use some other program. msieve is good.
[/QUOTE]

What's he up to now?

[QUOTE=3.14159;230236]What's he up to now?[/QUOTE]


there have only been 136 posts since the time you posted last it's not that hard to find out lol.

I've always thought of making a chess game on the web is it even possible in Pari ?

[QUOTE=science_man_88;230470]I've always thought of making a chess game on the web is it even possible in Pari ?[/QUOTE]

Pari is [url=http://en.wikipedia.org/wiki/Turing_completeness]Turing-complete[/url], so yes it can make a chess game. But it's not very well suited to the task!

This is old, but since you were interested:
[QUOTE=3.14159;227456]Again, if you're willing to fake ECM/SIQS/NFS data, show me how.[/QUOTE]

Pages 8-10 discuss faking ECM data:
[url]http://homepages.cwi.nl/~herman/Zimmermann.pdf[/url]

Note that step 0 is not required if the number doesn't need to be of special form (you can just create your own semiprime by multiplying random primes), and thus slide 11 doesn't apply.

[QUOTE=Charles;230792]This is old, but since you were interested:


Pages 8-10 discuss faking ECM data:
[url]http://homepages.cwi.nl/~herman/Zimmermann.pdf[/url]

Note that step 0 is not required if the number doesn't need to be of special form (you can just create your own semiprime by multiplying random primes), and thus slide 11 doesn't apply.[/QUOTE]

Next: How to fake SIQS data?

[QUOTE=3.14159;230795]Next: How to fake SIQS data?[/QUOTE]

Of course it's trivial to fake them at present since you're not asking for any data...

I don't know of an easy way at present, but let me think on it.

[QUOTE=CRGreathouse;230809]Of course it's trivial to fake them at present since you're not asking for any data...

I don't know of an easy way at present, but let me think on it.[/QUOTE]

You have any ideas? Anything at all?

[QUOTE=3.14159;232458]You have any ideas? Anything at all?[/QUOTE]

Yes -- but I'd prefer not to share until I get a working version.

[QUOTE=CRGreathouse;232477]Yes -- but I'd prefer not to share until I get a working version.[/QUOTE]

Well, you can notify me full speed ahead whenever you have those ideas.

[QUOTE=3.14159;232522]Well, you can notify me full speed ahead whenever you have those ideas.[/QUOTE]

I have the ideas now, but haven't worked through the math needed to implement them yet. I'll also need to find a system with a fast GCD -- I think GMP 5 has Moller's improvements coded in, so maybe that.

thought I'd put this here.
 

[CODE](13:30) gp > U(x,P,Q)= v=vector(x+1,n,0);v[1]=0;v[2]=1;for(y=3,x+1,v[y]=P*v[y-1]-Q*v[y-2]);print(v);
(13:33) gp > V(x,P,Q)= v=vector(x+1,n,0);v[1]=2;v[2]=P;for(y=3,x+1,v[y]=P*v[y-1]-Q*v[y-2]);print(v);[/CODE]

I deleted the tests of the stuff i pasted to save room. like them CRG and is there ever a third kind ? I know Mersenne numbers are U(P,Q) for P=3 Q=2 for this type but the V aren't quite the exponents I was wondering if we could ever make more kinds so that one may be a list of the exponents.

Yes, I like them, but I'd like it better with ";v" instead of ";print(v);". Always return, never print! With the return version you can do fun stuff like isprime(U(7,3,2)).

These are (homogeneous linear) recurrence relations (with constant coefficients); there are many more possibilities.

one thing I don't like is Wikipedia doesn't list it under lucas sequences I have a account just I forgot it all lol. maybe if someone remembers they can edit it lol.

What is "it"?

[QUOTE=CRGreathouse;233083]What is "it"?[/QUOTE]

Mersenne numbers it doesn't list them as a famous example in [url]http://en.wikipedia.org/wiki/Lucas_sequence[/url] but they are P=3 Q=2

That's hard because some definitions require them to have a prime exponent, in which case they're not recurrence relations.

[QUOTE=CRGreathouse;233087]That's hard because some definitions require them to have a prime exponent, in which case they're not recurrence relations.[/QUOTE]

couldn't there be a side note ? lol

now if only I could prove which primes using the Lucas formulas

[QUOTE=science_man_88;233088]couldn't there be a side note ? [/QUOTE]

You could add it in if you like. You don't even need to log into Wikipedia to make changes.

[QUOTE=CRGreathouse;233091]You could add it in if you like. You don't even need to log into Wikipedia to make changes.[/QUOTE]

added in I still want to figure out what the V sequence would represent.

[QUOTE=science_man_88;233094]added in I still want to figure out what the V sequence would represent.[/QUOTE]

2^n + 1, Sloane's [url=http://oeis.org/classic/A000051]A000051[/url].

[QUOTE=CRGreathouse;233095]2^n + 1, Sloane's [url=http://oeis.org/classic/A000051]A000051[/url].[/QUOTE]

if we could find a P,Q that would work for the exponents to each if possible the hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.

[QUOTE=science_man_88;233097]if we could find a P,Q that would work for the exponents to each if possible the hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.[/QUOTE]

What does "work" mean?

[QUOTE=CRGreathouse;233099]What does "work" mean?[/QUOTE]

succeed in creating the list in this case.

I found the p and q for 0 and positive integers numbers lol p=2 q= 1

[QUOTE=science_man_88;233130]succeed in creating the list in this case.

I found the p and q for 0 and positive integers numbers lol p=2 q= 1[/QUOTE]

OK, so you're saying
[INDENT]If we could find a P,Q that would succeed in creating the list for the exponents to each if possible the hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.[/INDENT]
So what is "the list", "succeed", and "started"?

[QUOTE=CRGreathouse;233173]OK, so you're saying
[INDENT]If we could find a P,Q that would succeed in creating the list for the exponents to each if possible the hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.[/INDENT]
So what is "the list", "succeed", and "started"?[/QUOTE]

the list = Mersenne prime exponents
succeed = successfulness in the creation
started = already underway

look if P and Q need to be integers then 3*3-2*2 = 9-4 = 5 the next one so since P needs to be 3 Q=2 is the only way to get them started but as we proved that's also the start of 2^n+1.

[QUOTE=science_man_88;233183]the list = Mersenne prime exponents
succeed = successfulness in the creation
started = already underway

look if P and Q need to be integers then 3*3-2*2 = 9-4 = 5 the next one so since P needs to be 3 Q=2 is the only way to get them started but as we proved that's also the start of 2^n+1.[/QUOTE]
[INDENT]If we could find a P,Q that would create the list of Mersenne exponents, ___. The hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.[/INDENT]
I can't entirely understand this, but it should be easy to show that there are no P, Q that yield the Mersenne prime exponents.

Ah yes, here we go. My program [url=http://oeis.org/wiki/User:Charles_R_Greathouse_IV/Pari#Recurrences]findrec[/url] says:
[code]Cannot be described by a homogeneous linear recurrence relation with 19 or fewer coefficients.[/code]
of the first 39 Mersenne exponents. So not only can you not find P, Q (a 2nd-order homogeneous linear recurrence relation with particular starting conditions), you can't find *any* HLRR of order 2, or even 3 through 19.

[QUOTE=CRGreathouse;233187][INDENT]If we could find a P,Q that would create the list of Mersenne exponents, ___. The hard part is if it has to be integer then the only way I see to get the Mersenne prime exponents started is 3,2 though I'm likely wrong.[/INDENT]
I can't entirely understand this, but it should be easy to show that there are no P, Q that yield the Mersenne prime exponents.

Ah yes, here we go. My program [url=http://oeis.org/wiki/User:Charles_R_Greathouse_IV/Pari#Recurrences]findrec[/url] says:
[code]Cannot be described by a homogeneous linear recurrence relation with 19 or fewer coefficients.[/code]
of the first 39 Mersenne exponents. So not only can you not find P, Q (a 2nd-order homogeneous linear recurrence relation with particular starting conditions), you can't find *any* HLRR of order 2, or even 3 through 19.[/QUOTE]

is there a formula for telling which in a lucas sequence are prime ?

[QUOTE=science_man_88;233214]is there a formula for telling which in a lucas sequence are prime ?[/QUOTE]

Sure, take any of the standard formulas for prime-testing and substitute the formula for a member of the Lucas sequence for the number.

But that would be very slow. Are you looking for an algorithm instead?

[QUOTE=CRGreathouse;233224]Sure, take any of the standard formulas for prime-testing and substitute the formula for a member of the Lucas sequence for the number.

But that would be very slow. Are you looking for an algorithm instead?[/QUOTE]

my latest idea if no prime finding formula that can work fast can be found is to create something that will feed the exponents into findrec in specific possible orders increasing that may find multiple lucas sequences I'm unsure right now though.

[QUOTE=science_man_88;233229]my latest idea if no prime finding formula that can work fast can be found is to create something that will feed the exponents into findrec in specific possible orders increasing that may find multiple lucas sequences I'm unsure right now though.[/QUOTE]

Well, you have the code... try it!

might need to rethink lol 39! possible for 39 and then 39!/2! for 38 etc. lol

something new ?
 

[CODE]for(i=2,#mersenne,print(isprime(floor(mersenne[i]^exp(1)/mersenne[i-1]))))[/CODE]

gives me false consistently, care to see if it's done already ? etc. I think someone else got in my account as I don't remember adding an image.

I was just pondering..

Imagine you choose a random 210-digit odd number to factor.

What are the odds that it has no factor ≤ 30 digits ?

Ex: [code]765223385698389926571242942756698700016608850019007376399764250461012579493486533505154279222065824923099207583225846866415511389377654307908279943765426193326813047827405221065725982667368472098358003394603467 = 77771442250823086981801725238987802825302970789365898005146018089 * 9839387872356080094484924867666017102610781172377705317233177547076926514496969631181594854116446488590453712956846616476891085418145551273102803[/code]

Roughly [TEX]\frac{2e^{-\gamma}}{\log10^{30}}\approx1.6\%[/TEX].

I find that exp(1) seems to work in numerous places I tried Pi in the same places and it didn't always turn out false.

[QUOTE=Charles;233663]Roughly [TEX]\frac{2e^{-\gamma}}{\log10^{30}}\approx1.6\%[/TEX].[/QUOTE]

Even though there are more potential factors ≥ 1000000000000000000000000000057 than there are of the lesser primes?

[QUOTE=3.14159;233800]Even though there are more potential factors ≥ 1000000000000000000000000000057 than there are of the lesser primes?[/QUOTE]

Correct. In fact, more than 30% of numbers have no prime factors greater than their square root. Think about it: there more than 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 [i]times[/i] as many primes between 1e10 and 1e100 as between 1 and 1e10, and yet 30% of the numbers near 1e100 use none of those many large primes.

[QUOTE=Charles;233803]Correct. In fact, more than 30% of numbers have no prime factors greater than their square root. Think about it: there more than 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 [i]times[/i] as many primes between 1e10 and 1e100 as between 1 and 1e10, and yet 30% of the numbers near 1e100 use none of those many large primes.[/QUOTE]

This statement can be distorted into some strange sort of crackpottery about some obscure pseudo-proof about some obscure pseudo-theorem.

[QUOTE=3.14159;233804]This statement can be distorted into some strange sort of crackpottery about some obscure pseudo-proof about some obscure pseudo-theorem.[/QUOTE]

I suppose any statement can, with enough (Levenshtein) effort...

Factors of 100-digit numbers;

8768935392562947618103320365401275447646525222169584791935566687049248245681935086698791112131179280 = 2[sup]4[/sup] * 5 * 7 * 229 * p95.

5013440241231690776250434282127412300350543560061260586054934713339836990739638416804096559649788785 = 5 * 13 * 9699973 * 1174300196009327 * 21278531935237641679 * 9306615253804741923704699323 * 34193173165859299637123433127.

Two can play that game.

1383597229331378476493295042730289164926892754069547350166289357488402239083316569155527338510512173 = 17 * 269 * 3204518441 * 697760349213711353909 * 5254579498151677786377319633 * 25751436085780192367390036356137118613

1226134517673268382787935537453090554039551405695194373164354832617671209370642217560662229657298470 = 2 * 3 * 5 * 131 * 619 * 628721 * 23852987 * 4724551547 * 10363456437839351504851859 * 686418473916303990455700991234048960669804471

4504635868071489691958247053230215472190113144381522888320546861068040654323199216497079846700145651 = 389 * 1249 * 9642973 * 2045233251289268647933076900071285541622713 * 470103901236160604912377322470525630823764859

1297098091109389427734623857619487484934106044886083070033690339492405858741869928141909886528666131 = 8317 * 11941 * 21327073 * 245809044309404623 * 59965612215340918627 * 41546460895888449456911011110597043412272105231

7448288592391672456339288547820229140957061588130168488233367000147015498831478910673540062648250669 = 7 * 23 * 61 * 827 * 7507 * 10477 * 19219 * 2307031 * 2497513 * 295393403176303499 * 69716460956822059780391 * 5112858994119557060987163901

2006581414691869533013736118005569210068797105484012314990944036018923405986462394666364748981293834 = 2 * 173 * 293 * 1069 * 4783 * 599093014896197532123744596093287948267 * 6461609735497599811892038514138831637953353686917

1029740705859616722557935599692863818295173314586300398254476931684298457572289392513244259443832990 = 2 * 5 * 73 * 199[SUP]2[/SUP] * 2803 * 17207646383 * 23341361566424419838977 * 108806050756253076134575189 * 290787051597685066876250670779

1479250575750629405583533546871938031514416077664003084605743742272187812384319917935052766003248180 = 2[SUP]2[/SUP] * 5 * 7 * 79 * 6871 * 25463 * 57047 * 61826203 * 358796701 * 799808381 * 1762119998489266483 * 428629288525583272511156276869590573727

2470685955216016137289558854053204746584489279198071269508371183297983597651454585258860897699629611 = 7 * 6229 * 346223 * 1077224275553 * 140671259709105592299571501927 * 1080024859552291303286190375729442638566243936449

2582787899792929109639761223985963200359419971336466309731940735382779345756386069144321580216562935 = 3[SUP]2[/SUP] * 5 * 7[SUP]2[/SUP] * 163 * 307 * 797 * 1723 * 71363 * 4865184453786808865917799950950769 * 49095101401206488590960022804247016107742250711

2491980943988512648154994168720542478864237721433866865571719881672143735588510153037196275622246583 = 3 * 37 * 1657 * 241489 * 303097 * 34848139 * 27108622699 * 85333941456585370142137 * 2296208868050293347620604354622609936964009

I have a challenge for you all;

k * n! + 1 = p[sup]2[/sup], where p is a prime integer, and where k ≤ n!

Ex: 5 * 8! + 1 = 449[sup]2[/sup]

Submissions: 256060*396^8560+1 (22242 digits)

Verification:

Primality testing 256060*396^8560+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Special modular reduction using zero-padded FFT length 10K on 256060*396^8560+1
Calling Brillhart-Lehmer-Selfridge with factored part 40.08%
256060*396^8560+1 is prime! (15.0315s+0.0242s)

Next collection project...

-Work on the factorial + 1 primes.

If you want, you can get started on the -1 analogues.

No, not n! + 1; k * n! + 1.

Aaaaannnnddd, I'll kick off at.. 18, where PFGW cannot trivially factor them anymore.

You can dive for the smaller ones. However, I will extend the range by a factor of 1.5

Up to 107..

Covered every prime under 170 or so digits..

Now, I have covered every prime under 206 digits..

better than what I already know ?
 

if I did my math correct only certain k work for a given property of the exponent to try for factors I was wondering if we could every limit this down enough to make it easier.

[QUOTE=science_man_88;234256]if I did my math correct only certain k work for a given property of the exponent to try for factors I was wondering if we could every limit this down enough to make it easier.[/QUOTE]

What k do you find as impossible?

Okay; Back to the old game. Listing factors of 10[sup]200![/sup] -1.

I submit; 110742186470530054291318013, 10000099999999989999899999000000000100001, and 15362898429170396757717888856328974146292496901433891193564055671816191643.

Also; Is 990001 a long prime? It has no repetition for at least 1/2 its decimal expansion.

And lastly, is there a method to determine whether or not a certain prime is a long prime?

[QUOTE=CRGreathouse;234269]What k do you find as impossible?[/QUOTE]

well for example if p is 3 mod 8 then

for 2*k*p+1 to be = +1-1 mod 8

well 2*3 +1 = 7 so to be 7 mod 8 you have k=1

the next one that has mod 8 7 is k=5

and they follow k=4x+1

and if i did the math correct for 1 mod 8 it becomes

k=4x

If you're making a reference to the earlier challenge;

Find me a number of the form k * n! + 1 which is a prime square;

It's very easy. Here's an example: 5 * 8! + 1 = 449 * 449.

Note; Don't depend on anything relating to polynomials; They're all irreducible.

An example of a k * b^n + 1 number that is a prime power; 545793 * 396^3 + 1 = 5821793 * 5821793.

no Pi I'm working on about Mersenne numbers.

[QUOTE=3.14159;234193]I have a challenge for you all;

k * n! + 1 = p[sup]2[/sup], where p is a prime integer, and where k ≤ n!

Ex: 5 * 8! + 1 = 449[sup]2[/sup][/QUOTE]

I find none with 8 < n < 10,000. I don't expect that any exist with larger n; the heuristic probability of finding one is less than 10^-17825.

[QUOTE=CRGreathouse;234282]I find none with 8 < n < 10,000. I don't expect that any exist with larger n; the heuristic probability of finding one is less than 10^-17825.[/QUOTE]

:orly owl: How did you arrive at those figures?

[QUOTE=3.14159;234285]How did you arrive at those figures?[/QUOTE]

For n <= 1e4: direct search; for n > 1e4: numerical integration.

The same technique suggests about 4.9 below 1e4, which is a pretty good estimate (there are 6).

What are the odds of a 210-digit odd number being a p105 * p105?

I'm guessing from 0.001% to 0.1%.

[QUOTE=3.14159;234294]What are the odds of a 210-digit odd number being a p105 * p105?[/QUOTE]

An easy upper bound is 0.00155%. A closer analysis gets 0.00127419%, though my calculations on the latter assume the Riemann hypothesis.

[QUOTE=CRGreathouse;234297]An easy upper bound is 0.00155%. A closer analysis gets 0.00127419%, though my calculations on the latter assume the Riemann hypothesis.[/QUOTE]

There's those unproven conjectures again..

[QUOTE=3.14159;234298]There's those unproven conjectures again..[/QUOTE]

Thus my first answer, which does not assume it. I could have given a better unconditional answer, but that would be too much work.

[URL="http://4.bp.blogspot.com/_rvR3ouziO8g/TMTL4iDv-uI/AAAAAAAAAuA/NPWgKtjtflE/s1600/rep.PNG"]ep[/URL]

the right mathematics can not cheat

[QUOTE=CRGreathouse;234299]Thus my first answer, which does not assume it. I could have given a better unconditional answer, but that would be too much work.[/QUOTE]

Hopefully it would be accurate..

[QUOTE=cmd;234301][URL="http://4.bp.blogspot.com/_rvR3ouziO8g/TMTL4iDv-uI/AAAAAAAAAuA/NPWgKtjtflE/s1600/rep.PNG"]ep[/URL][/QUOTE]

Hey, can you try spamming another thread, bozo?

I
 

[QUOTE=cmd;234301][URL="http://4.bp.blogspot.com/_rvR3ouziO8g/TMTL4iDv-uI/AAAAAAAAAuA/NPWgKtjtflE/s1600/rep.PNG"]ep[/URL]

the right mathematics can not cheat[/QUOTE]


[url]http://www.xkcd.com/747/[/url]

[url]http://www.xkcd.com/759/[/url]

[URL="http://4.bp.blogspot.com/_rvR3ouziO8g/TLxMpwU62TI/AAAAAAAAAtI/Vsj4-gSD0lM/s1600/cmdugo.PNG"]third element[/URL] of the first group,
advance of 30 and then tell us where to find last

[QUOTE=3.14159;234302]Hopefully it would be accurate..[/QUOTE]

It should be possible to calculate a reasonable range in which the answer lies. I'm not sure how precise it would be possible to make it.

I think I can sieve by hand for k * b^n ± 1 numbers; I just need to do some modular arithmetic to get rid of each prime p which divides a certain k;

Ex: k * 6^90 + 1;

Step 1; 6^90 mod 7 = 1.

Step 2; Multiply by 6; 6 * 6^90 = 6 mod 7;

6^91 + 1 mod 7 = 0.

7 divides k = {6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, ...}

Repeat for 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, and.. as far as you'll be willing to strain your hand..

Although even a computer is approx. 10^4 to 10^8 times faster than you.

I'll use that whenever working with no computer assistance; Unless there's an easier method to go by.

[CODE](10:15) gp > for(i=1,#mersenne,print(isprime(floor(prime(i)^(2*Pi)))))
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(10:16) gp > for(i=1,#mersenne,print(isprime(floor(mersenne[i]^(2*Pi)))))
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(10:16) gp >[/CODE]

can we use this fact that all the Mersenne exponents up to #39 work but not all primes up to prime 39 could this help narrow down exponents if used ?

though it may be doubtful these continue and it took me a long time to find a 1 in the prime[i]^(5*Pi) one and they seem to work for Mersenne exponents so far as well.

[QUOTE=science_man_88;235192]can we use this fact that all the Mersenne exponents up to #39 work but not all primes up to prime 39 could this help narrow down exponents if used ?[/QUOTE]

No.

First, we don't know that the pattern will continue -- in fact I doubt it does. Second, there are way too many primes for which this will work -- heuristically, [TEX]\frac{n}{2\pi\log^2n}[/TEX] up to n.

But if you think you're onto something, try testing
49982629, 49985387, 49985521, 49986509, 49986737, 49987409, 49987759, 49988693, 49989491, ...
for which the relation holds and are toward the small end of the unassigned range. (I don't know which of these may already be taken, possibly all.)

[QUOTE=CRGreathouse;235200]No.

First, we don't know that the pattern will continue -- in fact I doubt it does. Second, there are way too many primes for which this will work -- heuristically, [TEX]\frac{n}{2\pi\log^2n}[/TEX] up to n.

But if you think you're onto something, try testing
49982629, 49985387, 49985521, 49986509, 49986737, 49987409, 49987759, 49988693, 49989491, ...
for which the relation holds and are toward the small end of the unassigned range. (I don't know which of these may already be taken, possibly all.)[/QUOTE]

okay but how many of those fit for the 5*Pi one ? or the sequence of x*Pi that are :

2 5 8 10 11 12 14 15 18 all Mersenne exponents so far tested by me work to be 0 ( false) for all of these but not every prime will work in these relations.

[QUOTE=CRGreathouse;235200]No.

First, we don't know that the pattern will continue -- in fact I doubt it does. Second, there are way too many primes for which this will work -- heuristically, [TEX]\frac{n}{2\pi\log^2n}[/TEX] up to n.

But if you think you're onto something, try testing
49982629, 49985387, 49985521, 49986509, 49986737, 49987409, 49987759, 49988693, 49989491, ...
for which the relation holds and are toward the small end of the unassigned range. (I don't know which of these may already be taken, possibly all.)[/QUOTE]

okay but how many of those fit for the 5*Pi one ? or the sequence of x*Pi that are :

2 5 8 10 11 12 14 15 18 all Mersenne exponents so far tested by me work to be 0 ( false) for all of these but not every prime will work in these relations.

f each combo of isprime !isprime etc. knocks out a prime when the range is up to 1,000,000 then this can knock out over 60% of prime exponents in the range (though it might slump lower) without a single LL test being done. if confirmed it would help a lot I think. not quite sure anymore got to figure it out there are 623530 combinations of taking the numbers i've found to work so far off isprime I believe if so then if each one eliminated just one exponent we could eliminate 623530 primes at least from ever needing LL test.

[QUOTE=science_man_88;235209]okay but how many of those fit for the 5*Pi one ?[/QUOTE]

You have Pari/GP, feel free to test. Heuristically, if I understand what you're asking, about [TEX]\frac{x}{10\pi^2\log^4x}[/TEX] numbers up to x will have this property and the other.

But I emphasize that this is likely to remove as many (as a fraction) of future Mersenne exponents as Mersenne non-exponents.

Edit: I can't decide on what exponent to use in the denominator, but not less than 2 or more than 4.

[QUOTE=CRGreathouse;235222]You have Pari/GP, feel free to test. Heuristically, if I understand what you're asking, about [TEX]\frac{x}{10\pi^2\log^4x}[/TEX] numbers up to x will have this property and the other.

But I emphasize that this is likely to remove as many (as a fraction) of future Mersenne exponents as Mersenne non-exponents.[/QUOTE]

yeah scrap 18 lol for use for Mersenne exponents yeah well it's based on properties of the rest of them to predict the next lol.

Here's a nice challenge for you all;

Express the number 988741313296003[sup]47[/sup] as a sum of prime powers, which meet these conditions:

1. No more than 25 primes may be used.

Ex: 2[sup]30[/sup] + 3[sup]18[/sup] + 5[sup]12[/sup] + 19[sup]6[/sup] + 293[sup]3[/sup] + 577[sup]3[/sup] + 191[sup]3[/sup] + 89[sup]3[/sup] + 223[sup]2[/sup] + 37[sup]2[/sup] + 11[sup]2[/sup] + 73[sup]1[/sup] + 2[sup]1[/sup] = 7[sup]11[/sup]

Or, for a harder one; No primes smaller than 250000 may be used, no primes may be used more than once, all exponents must be prime and used once and only once, and no more than 30 primes may be used.

[QUOTE=3.14159;236590]Here's a nice challenge for you all;

Express the number 988741313296003[sup]47[/sup] as a sum of prime powers, which meet these conditions:

1. No more than 25 primes may be used.

Ex: 2[sup]30[/sup] + 3[sup]18[/sup] + 5[sup]12[/sup] + 19[sup]6[/sup] + 293[sup]3[/sup] + 577[sup]3[/sup] + 191[sup]3[/sup] + 89[sup]3[/sup] + 223[sup]2[/sup] + 37[sup]2[/sup] + 11[sup]2[/sup] + 73[sup]1[/sup] + 2[sup]1[/sup] = 7[sup]11[/sup]

Or, for a harder one; No primes smaller than 250000 may be used, no primes may be used more than once, all exponents must be prime and used once and only once, and no more than 30 primes may be used.[/QUOTE]

if only I knew how to implement the four squares theorem, or the theorem that states every number can be expressed as a sum of 19 fourth powers in such a way to use it to figure out a way to represent the base with prime powers I could maybe do it lol.

pi your base is prime hence no configuring needs to be done for 1. as for 2 I'll try it out maybe lol. oh wait if 1 prime base and exponent pair is needed you gave it stating the question my work is done. okay maybe not as the 250000 part you're saved from completely being destroyed by me.

[QUOTE=3.14159;236590]Express the number 988741313296003[sup]47[/sup] as a sum of prime powers, which meet these conditions:

1. No more than 25 primes may be used.

Ex: 2[sup]30[/sup] + 3[sup]18[/sup] + 5[sup]12[/sup] + 19[sup]6[/sup] + 293[sup]3[/sup] + 577[sup]3[/sup] + 191[sup]3[/sup] + 89[sup]3[/sup] + 223[sup]2[/sup] + 37[sup]2[/sup] + 11[sup]2[/sup] + 73[sup]1[/sup] + 2[sup]1[/sup] = 7[sup]11[/sup]

Or, for a harder one; No primes smaller than 250000 may be used, no primes may be used more than once, all exponents must be prime and used once and only once, and no more than 30 primes may be used.[/QUOTE]

Well, 988741313296003[sup]47[/sup] is a solution to both, so I imagine you intended to disallow that.

For the first, the simplest remaining solution is x[sup]1[/sup] + 5441[sup]1[/sup] + 3[sup]1[/sup].

The second seems hard. I assume that the size limit on primes does not apply to exponents, or else the problem has no solutions. It's not clear whether you're limiting the number of summands to 15 or 30.

[QUOTE=science_man_88;236597]pi your base is prime hence no configuring needs to be done for 1.[/QUOTE]

You beat me to it!

I think I might have been able to phrase that a bit better..

[QUOTE=CRGreathouse;236604]You beat me to it![/QUOTE]

I was in the thread just after he posted it and and had pari open so I tried it out lol.

Okay; Try answering this;

Can every integer > 10[sup]6[/sup] be expressed as a sum of prime squares?

Example: 1802042 = 1009^2 + 877^2 + 113^2 + 43^2 + 13^2 + (3^2) * 5

Another; 545014337 = 23117^2 + 3167^2 + 761^2 + 97^2+ 11^2 + (2^2)*18 + (3^2)*4

[QUOTE=3.14159;236735]Can every integer > 10[sup]6[/sup] be expressed as a sum of prime squares?

Example: 1802042 = 1009^2 + 877^2 + 113^2 + 43^2 + 13^2 + (3^2) * 5

Another; 545014337 = 23117^2 + 3167^2 + 761^2 + 97^2+ 11^2 + (2^2)*18 + (3^2)*4[/QUOTE]

Yes. In fact, this holds for every integer greater than 23. (To prove this it suffices to prove it for {24, 25, 26, 27}. Alternately, a method of Sylvester (1884) allows a one-step proof for n > 35.)

If you meant "distinct prime squares", then the answer is still yes, though I can't give an elementary proof. (A proof is not difficult with a little bit of computer power plus a weak bound on the growth of primes, e.g., p[sub]n+1[/sub] < 1.4p[sub]n[/sub].) The largest number that can't be so expressed is 17,163.

Part 2 of; Factoring random 100-digit numbers;

1000309121763716018192725963203643789700149527936906867199570524663892824104343656283438202733442867 = 3 * 827 * 72245231489 * 15956241661 * p32 * p43

5368323813249515329098296538291057500943044787302529648340980984764463865258985384399553193181390034 = 2 * 3[sup]2[/sup] * 17 * 29 * 113 * 1218 * 51503 * 141397 * p81

8013176147722647871956980350400065506099323485506514645505443741520820451558115945129175079407373895 = 5 * p22* p33* p46

1152443181169586383493536135294025497382757172618548038480830431161568631322294266650168720443874455 = 5 * p99

7666551255558602189678572939665486145558049206656581150748267177043910125431974118219650818960910581 = 59 * p99

2562808938752647864331131924722585579690624807298066926175303729176944775263748281662129402089103030 = 2 * 5 * 22153 * 19445759572794150941 * p75

8199342366679060827617296146239673154845942561339379994399286622024273611547154072068685152481812583 = 7 * 163 * 69149 * 133831 * p87

3817078147299184807266522855781210143244534280399726146655814305698076033398396433478794563004496650 = 2 * 5[sup]2[/sup] * 7193 * p21 * p74

2483514470656621527509543660684300711436807489222617550372160227312222630960322334656545109061468245 = 5 * 7 * 53 * p20 * p77.

5342453719772700595767968882054794877986643318570295675203507607631997227930905300848205513566413849 = 129749 * p95.

There you have it, ten 100-digit numbers, completely factored.

Pfft, here's 100.

Oops, the forum won't let me post that many characters. OK, here's the amount I can fit into a post.

[code]3828341080244400637832496477959805704164073468183074462634180957833651864984723682138669035433026500 = 2 * 2 * 3 * 5 * 5 * 5 * 7 * 19 * 19 * 23 * 37 * 53 * 241 * 1181 * 5649913 * 36081377 * 8581181173 * 47840420649221 * 56402348128571 * p32
1588660322983396302885796245830814329302488843364165698215578492464397730424401789229489704601075756 = 2 * 2 * 3 * 7 * 13 * 23 * 47 * 61 * 181 * 15031 * 526483 * 927323 * 1148527 * 406979061818023 * 81538217474440208479553 * p30
1021031048497837374483531447808891337935989532331588576367878602517111431033593224173329292881512173 = 31 * 367 * 30809 * 5668727 * 908392703 * 1028314699 * 2197843586849586383840467 * p42
7086422853097868451711345316509410180470376785600011857778722004533443743419826800885302936650498470 = 2 * 5 * 13 * 19 * 29 * 431 * 587 * 691 * 2767 * 942187 * 1893539 * 2597779 * 15624896897 * 1610494874908351 * 3845935534903586029 * p21
4318456126006444342857842557107114096633481415982568386769884274414806491142401044261345969736720694 = 2 * 3 * 7 * 11 * 13 * 23 * 31159 * 302053 * 30936197387 * 4176718791229 * 847849456387121 * 2254092997188947242409 * p26
1109587849660304353836201770893761411871650688020685001076825645933095103648972795052962839552595690 = 2 * 3 * 3 * 5 * 7 * 23 * 31 * 67 * 163 * 85632998041 * 17249535181483504313 * 394021370780626750477817951 * p33
2979941477970790621634249796909439789746784691205449811130412558873271515382744651517007594645175894 = 2 * 3 * 11 * 13 * 19 * 97 * 1637 * 706109 * 60265063 * 59542983281711851 * 6457175558840031600628637783 * p32
6420726987540740841706278258860741277295366130264261913963063511707914602848605467111390751821088121 = 3 * 7 * 97 * 1481121042485345479125151 * 176615229066307992468364472796139 * p41
9816138939953536743143344071912358383004405759221258712071041553368290365584815105024598168207976103 = 839 * 10399 * 4264780100791321147 * 8561427410274222444303852083 * p47
2268168812216914591854154823153328155076179646049376828112834528320073183563365268538263313054628020 = 2 * 2 * 3 * 3 * 5 * 7 * 7 * 23 * 31 * 31 * 37 * 61 * 73 * 479 * 523 * 1486553277901 * 2312246873086067716693912462289027 * p35
2175187002194884841675973456689521692815926892066030053856510088603896308976058660302016139102867286 = 2 * 17 * 43 * 199 * 317 * 17207 * 595611953 * 374487601672675901611 * 120397871923043244929719 * p35
1964666465908214969730131549122817558003317961763724416749317778247127635667033822930173507293655037 = 1409 * 2621779 * 572550962994499639130106764801815763499954743 * p45
2109835988366518432588338355806050669474506758795640108970282822027494388216935165502614463059840500 = 2 * 2 * 3 * 3 * 5 * 5 * 5 * 13 * 17 * 53 * 53 * 89 * 241 * 279464069 * 13390332799 * 13566564771765494922641095631 * p39
4170436258982883679101363076894140796229193134975892831177906516068402234097418053129515314672448236 = 2 * 2 * 17 * 2069 * 131381 * 147541 * 392669 * 1385569 * 8654839 * 1817213413 * 641105533842841 * 22777900103786411 * p26
1112745709173761896559835695316280609014296134651327666778801370828688600198148161143564180857076834 = 2 * 3 * 7 * 11 * 31 * 47 * 317 * 19381 * 182617 * 304849 * 589163046401 * 490279242759107592236447059 * p38
6754467728048555661382638518307179841909700849996023430663298769830785246937785624396484840143801820 = 2 * 2 * 5 * 13 * 19 * 23 * 29 * 83 * 97 * 2029 * 102139 * 175919 * 28920977 * 37222903 * 3325507556831 * 113026544095967 * p35
9290303292654677796208450313997479786862212010805900056412444714830815535310109739094853648011955880 = 2 * 2 * 2 * 3 * 5 * 19 * 19 * 53 * 139 * 149 * 2633 * 254557 * 7518315653802366207934294760412271 * p47
1519045907326352179272495192012691255110343871460740344723603485053123113376600850827082587042722632 = 2 * 2 * 2 * 3 * 3 * 3 * 7 * 7 * 7 * 7 * 11 * 13 * 19 * 23 * 73 * 953 * 8039 * 191531 * 650543 * 966923 * 2910837913 * 4712082339471169543 * p35
2297494279297747651318689357388091486187167887028748530651383542166405596399953892245633133215771067 = 3 * 19 * 1129 * 4909 * 45281 * 95483 * 350573551 * 185546423887211 * 5000505715976357 * p43
6916716710106028694497858360968629919200917839743569117166581989808297983518264029006195056926115135 = 3 * 5 * 7 * 313 * 599 * 701 * 887 * 117331 * 4657853 * 8510059 * 3519961031 * 34327729151 * 346541139438472730910749 * p25
4020646510893532589104720126916970485917983107805030705775412066914614392060823406100284080206300400 = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 5 * 7 * 7 * 7 * 13 * 31 * 31 * 43 * 67 * 349 * 33623 * 102679 * 142159 * 229717 * 10307723 * 338788421 * 1427600611 * p39
7172656039606291939374148166929125456455438300989478240428912337825056979098115354643207581855835285 = 3 * 3 * 5 * 7 * 13 * 19 * 37 * 61 * 487 * 1121941560287 * 121409626878654282391474088573 * p48
2528919604826729658531053105058688132708635900874626524727189778747923659829992520183389956689288493 = 7 * 19 * 37 * 127 * 887 * 2347 * 149551999 * 140397312799 * 46097360176928611953517733 * p43
4780580308922684791934717889134818971136050278964890263719923034086360164542392367179511433499581230 = 2 * 3 * 5 * 11 * 47 * 107 * 199 * 3559 * 101399 * 226307 * 1130190559 * 12732542650017211591 * 130385357953530112759 * p29
1771581299718052268780183255753188184264364543991851960093639623289282075054030826702670725925580126 = 2 * 3 * 7 * 23 * 73 * 1621 * 440681 * 57252901 * 9424402487 * 555590203710791497 * 1664351435128005575192027 * p26
1826607301881605308867617623466131387321007525492908119055581832481636954149723135560049573051779689 = 3 * 7 * 8731 * 11971 * 86423 * 133697 * 6175693 * 3399940349409841 * 228101887805109423382486397 * p32
5012088009277640314402761559890041839759396057018545897258241994190447331558998944288170990344028751 = 2917 * 1757549 * 85744849549 * 77390355798296212981 * 117857840533006506831097561 * p34
1118682071008816168222725102230550120702126996309533995481998016819660387118847018742147190609638748 = 2 * 2 * 3 * 7 * 7 * 7 * 11 * 31 * 31 * 15107 * 66041 * 3629831 * 32303143 * 3644664791 * 3669739756759 * 9842283515159 * p34
3409719665791495912088557548714548870555928299445478456550560379913289306779291543643147189360470373 = 13 * 63079 * 1897177 * 6700089011507840929048948172217533113246337 * p45
1495013387078444830971209867702573522460009718985740283372631172961535442849970157536302235575256418 = 2 * 3 * 7 * 7 * 7 * 23 * 43 * 683 * 1069 * 11941 * 79943 * 174767 * 3055399 * 15229428191 * 97167102001 * 342494678174970979223 * p25
4525811937133892039004695583122389645196403182725701117694456147561445885523093800196418939098804213 = 3 * 7 * 127 * 181 * 971 * 1249 * 2417 * 938338889863 * 1435055134111 * 14576177493396886245107 * p39
2073311625512958451771092516352979522018668970964146344058841979417884026793493734398961719458991714 = 2 * 7 * 83 * 683 * 2927 * 2853337 * 14091887522033 * 229463148114318047593 * 119858213886351109516757 * p27
2898380564764331740650976346727898110985597681040329046593860520663948977367124731596264106185936093 = 3 * 3 * 101 * 151 * 271 * 1373 * 2153 * 17239 * 81239 * 2773871 * 7857481 * 899962939 * 82370557151 * 1455761853057791 * p28
3198477157946413712356473944979032360351567621787560078910437352396253167806937403734139451063110790 = 2 * 5 * 31 * 1543 * 31643 * 16930549 * 4315089697 * 147331735717257383607543819531715979 * p38
3122316024826623570999390780435061018523720250135414481535910292606995414401907413802207852764507150 = 2 * 5 * 5 * 13 * 53 * 9941 * 1095839 * 1436933 * 3566369 * 116278069 * 704550047 * 3739896647 * p46
1150279127692742245826975952975244459786635097720009368565532437170228458297750980752159689159559618 = 2 * 3 * 7 * 43 * 211 * 74843 * 487703 * 179282303 * 5191468757 * 279705792443006937347 * p45
3476994682280465721380107970080979643964773181753875578457560808630413498598886246958613562097879220 = 2 * 2 * 3 * 5 * 13 * 191 * 5912088495769122635780489 * 7782116779738216343902940458011407 * p36
2396930940518056525873755201487754594494952937746349126647824329949425055589025208870046787995076360 = 2 * 2 * 2 * 3 * 3 * 3 * 5 * 17 * 1361 * 13049 * 117243743 * 3537179857 * 13631427749 * 24865862748311267 * 83386179669921800083 * p24
5059444662280282391945087488109772319758228923707259715338382813207657103741168542070840844904146220 = 2 * 2 * 3 * 3 * 5 * 7 * 23 * 29 * 79 * 101 * 113 * 743 * 4831 * 2962207 * 4864957 * 15387939349 * 30166451450413001 * 22509974642730605057 * p23
9047865069205221785796363229531231728097227782231483365308238450563871586695628928251384364335276289 = 3 * 7 * 11 * 13 * 37 * 41 * 1327 * 1361 * 4243 * 29153 * 47513 * 9410983 * 55253941 * 219090281 * 1296987047 * 8269682713 * 84184045307 * p22
4818738698309238680979976982780739771434590565295500757251227197031479934194199257557409828463541162 = 2 * 31 * 79 * 379 * 25868233 * 64475011693 * 10929957278893 * 26357427070885165852945211371 * p34
7569920572151463447485708659596986502379885621035578545383014239308628711137659305904538890841093447 = 3 * 7 * 13 * 37 * 47 * 53 * 149 * 401 * 3457 * 34673 * 2494512199 * 13965069563 * 32420504751992087 * p44
7384425977124614373037190268769494322670515018022021984540726867123494391479847118318115045923964027 = 11 * 103 * 241 * 719 * 148709668309 * 51020820806647045557427500151889591 * p46
1460986172293537551768238256080060645982228305051951190204471703642679618463315789617586532311138454 = 2 * 3 * 31 * 47 * 673 * 1051 * 29478523 * 2049939233 * 120265998807826733 * 292426016421719911 * p39[/code]

Were those random or not?

Oh, wait, it's stupid to ask that question, probably pseudorandom and therefore nonrandom.