Let's see how you loophole your way out of that one!

[QUOTE=3.14159;227357]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.[/QUOTE]

No thanks, my cores are busy extending several sequences relating to pseudoprimes and large prime gaps. I do enjoy finding trivial examples, though.

[QUOTE=3.14159;227358]Let's see how you loophole your way out of that one![/QUOTE]

One possibility: go through 20-digit primes, finding k and n values that give a (521 to 540)-digit multiple of that prime, checking if the cofactor is a 2-pseudoprime. This should be faster than actually factoring them out. But I'm not going to do this search because it's just as silly as it sounds.

[QUOTE=CRGreathouse]No thanks, my cores are busy extending several sequences relating to pseudoprimes and large prime gaps. I do enjoy finding trivial examples, though.
[/QUOTE]

Ah, but you could have loopholed your way out of this one, as I made no digit requirement. It's as easy as:

1230021358933920151693112553959118376943838 * 400[sup]35[/sup] + 1 = 8826736195555687888973540222790863753879 * 1645175382483954577575076564167802963870136473616621363381331706098141318612978687213529959719.

And, voila! You would have won, yet again.

[QUOTE=3.14159;227354]Never said the divisors had to be small[/QUOTE]

I suppose you could have cofactor (small factor) and cofactor (large factor) categories. It would be interesting to properly classify the difficulty of finding (1) a general and (2) a special-form cofactor based on the size of both the small and large factor. The main difficulty is the latter, of course, but the former plays a part too.

[QUOTE=CRGreathouse]One possibility: go through 20-digit primes, finding k and n values that give a (521 to 540)-digit multiple of that prime, checking if the cofactor is a 2-pseudoprime. This should be faster than actually factoring them out. But I'm not going to do this search because it's just as silly as it sounds.
[/QUOTE]

See above

[QUOTE=3.14159;227363]See above[/QUOTE]

I beat you to it. :razz:

[QUOTE=CRGreathouse]I suppose you could have cofactor (small factor) and cofactor (large factor) categories. It would be interesting to properly classify the difficulty of finding (1) a general and (2) a special-form cofactor based on the size of both the small and large factor. The main difficulty is the latter, of course, but the former plays a part too.
[/QUOTE]

A special-form cofactor in a special-form number? Please, this only happens in Mersenne numbers. If you wanted a special-form factor, you would have to directly rig it to do so.

[QUOTE=3.14159;227357]the digits must follow a random distribution that is [B]statistically likely[/B][/QUOTE]

How would you define this, I wonder?

[QUOTE=CRGreathouse]How would you define this, I wonder?
[/QUOTE]

No predictable sequences. No patterns in any section of digits.

[QUOTE=3.14159;227366]A special-form cofactor in a special-form number? Please, this only happens in Mersenne numbers. If you wanted a special-form factor, you would have to directly rig it to do so.[/QUOTE]

There are many numbers that have factors of special form, not just Mersennes. (And [i]of course[/i] I'm rigging it! Since when have I not?) But Mersenne numbers would be a great choice, probably where I'd start. You can trial-divide much more easily when you skip over 6p numbers at a stroke...

[QUOTE=3.14159;227368]No predictable sequences. No patterns in any section of digits.[/QUOTE]

Suppose this was for a competition and you needed a definition that could be programmed into a computer or written in a rulebook so there could be no disputes. How would you define it?

I'm honestly curious.

[QUOTE=CRGreathouse]There are many numbers that have factors of special form, not just Mersennes. (And of course I'm rigging it! Since when have I not?) But Mersenne numbers would be a great choice, probably where I'd start. You can trial-divide much more easily when you skip over 6p numbers at a stroke...
[/QUOTE]

Also: I don't count those because it's too easy to find a prime using those numbers. Far too easy. I require the factor to be a general prime. Or I could divide Cofactor into Special Cofactor and General Cofactor. Yeah, I'll do that.

[QUOTE=CRGreathouse]Suppose this was for a competition and you needed a definition that could be programmed into a computer or written in a rulebook so there could be no disputes. How would you define it?
[/QUOTE]

I just did. See my previous post.

[QUOTE=3.14159;227371]Also: I don't count those because it's too easy to find a prime using those numbers. Far too easy. I require the factor to be a general prime.[/QUOTE]

Why don't you re-post your list of prime records you're looking for, with these constraints and requirements listed -- no Mersenne cofactors, no small cofactors, the splitting of cofactors into 2 or 4 categories, only 'statistically likely'/un-'predictable sequences' (whatever that means), etc.

[QUOTE=CRGreathouse]Why don't you re-post your list of prime records you're looking for, with these constraints and requirements listed -- no Mersenne cofactors, no small cofactors, the splitting of cofactors into 2 or 4 categories, only 'statistically likely'/un-'predictable sequences' (whatever that means), etc.
[/QUOTE]

Split Cofactor into Special Cofactor and General Cofactor. Also: A Mersenne number for a cofactor? I never held any restriction against that. In fact, that is welcomed into Special Cofactor.

[QUOTE=3.14159;227371]I just did. See my previous post.[/QUOTE]
[which was]
[QUOTE=3.14159;227368]No predictable sequences. No patterns in any section of digits.[/QUOTE]

That's still ambiguous. I was looking for a mathematical definition, something a computer could give a firm "yes" or "no" to. I wouldn't want to search for something that I thought was fine only to have you tell me that it's not random enough for you for some reason I couldn't have guessed.

[QUOTE=CRGreathouse]That's still ambiguous. I was looking for a mathematical definition, something a computer could give a firm "yes" or "no" to. I wouldn't want to search for something that I thought was fine only to have you tell me that it's not random enough for you for some reason I couldn't have guessed.
[/QUOTE]

A computer is an idiot and is unable to understand anything. What's so ambiguous about randomly-generated numbers? (Well, at least pseudorandomly.)

Updated list:
1. [B]Proths[/B], where b is 2.
2. [B]Generalized Proths[/B], where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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 primorial 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]Special Cofactor[/B], where the prime cofactor is of one of the forms used in this list.
18. [B]General Cofactor[/B], where the prime cofactor is not of a special form.
19. [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.
20. [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.

Oh: Mersennes are not in the list. Ah, well. Mersenne number cofactors are classified as General Cofactor, then.

[QUOTE=3.14159;227375]What's so ambiguous about randomly-generated numbers?[/QUOTE]

Because if I hand you a number, you don't know where it came from. At this point I'm not even sure what you intend, let alone how precisely to define it. You've described it differently each time: "statistically likely", "No predictable sequences. No patterns in any section of digits.", and now "randomly-generated".

[QUOTE=CRGreathouse]Because if I hand you a number, you don't know where it came from. At this point I'm not even sure what you intend, let alone how precisely to define it. You've described it differently each time: "statistically likely", "No predictable sequences. No patterns in any section of digits.", and now "randomly-generated".
[/QUOTE]

No patterns in any section of digits.
No predictable sequences.
Statistically likely.

Don't those stand out to you as the characteristics of a randomly-chosen number?

Also: View above, I posted the updated list.

I can break or bend at least #3, #4, #5, #6, #7, and #19 on your newest list. (Not that you need to care about what trivialities I can dredge up, of course.) I'm still fundamentally opposed to #20: why encourage people to harm the environment by using more electricity than needed to prove a prime? And why wouldn't they just cheat and use a newer program?

For #18, I take it that this means that neither factor is one of #1-#15?

#16 looks cool.

[QUOTE=3.14159;227378]No patterns in any section of digits.
No predictable sequences.
Statistically likely.

Don't those stand out to you as the characteristics of a randomly-chosen number?[/QUOTE]

Is 4584747435685437658437 such a number? How do you know? What about [code]17935520800145983429475264219083132046283745750116734433889605552003293726651449435812688477850840442133972456012956887645193084972760632253129406[/code]
? What about 2665124305184977723311826852179758605262374940671173169012746835211677949?

[QUOTE=3.14159;227380]A demonstration of how easy it is to find a large special-form cofactor: [/QUOTE]

I think I showed that in #969 and #973 well enough. :smile:

[QUOTE=3.14159;227380]Yes, yes, fuse them into Generalized. But Generalized is not inclusive to factorial/primorial/prime numbers. Updating.
Same for the rest.[/QUOTE]

I can still bend/break #3, #4, #5, #6, #7, and #19. #8 isn't hard either; #2 is largely but not entirely fixed by the change. #19 has at least two trivial vulnerabilities, but it almost doesn't count since the worst vulnerability you already know about and have declined to change/fix.

[QUOTE=CRGreathouse]I can still bend/break #3, #4, #5, #6, #7, and #19. #8 isn't hard either; #2 is largely but not entirely fixed by the change. #19 has at least two trivial vulnerabilities, but it almost doesn't count since the worst vulnerability you already know about and have declined to change/fix.
[/QUOTE]

3: Nothing wrong with it: The only thing you keep complaining about is fusing it into Generalized.
4: Same as 3.
5: Same as 4.
6: Yes, this does not stand to the true definition of primorial prime.
7: Same as 6.
19: 10^k + c and something else.

[QUOTE=3.14159;227384]3: Nothing wrong with it: The only thing you keep complaining about is fusing it into Generalized.[/QUOTE]

Actually, I've never complained about that or found it to be a vulnerability. I have a better break in mind.

[QUOTE=3.14159;227384]19: 10^k + c and something else.[/QUOTE]

Why would I want to use that?


Any response to #993?

[QUOTE=CRGreathouse]Actually, I've never complained about that or found it to be a vulnerability. I have a better break in mind.
[/QUOTE]

Invalid, same restrictions apply to those as they do to proths, k < b[sup]n[/sup].

[QUOTE=CRGreathouse]Why would I want to use that?
[/QUOTE]

Every number greater than 10^1999+100 can be expressed as such. Is this ever going to end?

[QUOTE=CRGreathouse]Any response to #993?
[/QUOTE]

Nothing to respond to in Post 993.

If there is any weird/distorted/strange crap, ask mods to remove it, please. The network is failing me, and my posts tend to be mangled when network failure occurs during me typing a post.

Here's one for categories #2 and #19:

[I]Start: For n=18778 to 18778, For k=494 to 494 step 2, k*288^n+1.[/I]
[I]494*288^18778 + 1 may be prime. (a = 2)[/I]
[I]494*288^18778 + 1 is prime! (a = 5) [46186 digits][/I]

Found with PFGW as a PRP, proved with Proth.exe. (Actually, it was proved first with PFGW...it was found by a script for searching generalized Sieprinski/Riesel conjectures that we use at the Conjectures 'R Us project. The script automatically proves PRPs upon finding them as such, so I didn't have the opportunity to [I]first[/I] prove it with Proth.exe. Does it still count? :smile:)

Edit: sorry, was composing this before you posted your latest requirements for the various categories; scratch #19 and make it #20.

Great. Now that the previous issue has been cleared up.. Post 1000 of the thread! :party:

[QUOTE]Found with PFGW as a PRP, proved with Proth.exe. (Actually, it was proved first with PFGW...it was found by a script for searching generalized Sieprinski/Riesel conjectures that we use at the Conjectures 'R Us project. The script automatically proves PRPs upon finding them as such, so I didn't have the opportunity to first prove it with Proth.exe. Does it still count? :smile:)

Edit: sorry, was composing this before you posted your latest requirements for the various categories; scratch #19 and make it #20.[/QUOTE]

Obsolete tech: No, because the proof was already done by PFGW. You techically used newer tech to do it first. The proof is to be done as follows: Grab the PRP, and have Proth.exe prove it prime. No #20 award for you, but you still get a spot for #2.

Also: Reached a nice p740 special cofactor. *Gives a #17 to self.*

[QUOTE=3.14159;227386]Nothing to respond to in Post 993.[/QUOTE]

Sorry, you deleted a post and it's now [url=http://mersenneforum.org/showpost.php?p=227381&postcount=992]#992[/url]. I had two questions about each of three different numbers there.

[QUOTE=CRGreathouse]Sorry, you deleted a post and it's now #992. I had two questions about each of three different numbers there.
[/QUOTE]

Settling this quickly:

The first number, you typed, the last two, randomly-generated.

[QUOTE=3.14159;227376]Updated list:
1. [B]Proths[/B], where b is 2.
2. [B]Generalized Proths[/B], where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
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 primorial 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]Special Cofactor[/B], where the prime cofactor is of one of the forms used in this list.
18. [B]General Cofactor[/B], where the prime cofactor is not of a special form.
19. [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.
20. [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.

Oh: Mersennes are not in the list. Ah, well. Mersenne number cofactors are classified as General Cofactor, then.[/QUOTE]

so many wasted numbers generalized with all exponents takes #2 to mean #1+#3+#4+#5, #8= #9+#10+#11. so generalized could 7 of them off the list at least. oh and #17 is a subtype of 18 if I did my logic correctly. #12 purposely stops it from being #13+#14+#15

[QUOTE=science_man_88]so many wasted numbers generalized with all exponents takes #2 to mean #1+#3+#4+#5, #8= #9+#10+#11. so generalized could 7 of them off the list at least. oh and #17 is a subtype of 18 if I did my logic correctly. #12 purposely stops it from being #13+#14+#15
[/QUOTE]

You can tear your heart out with as many complaints as you'd like, but that list is going to remain the way it is.

The only changes that are necessary are the changes to #19.

General arithmetic progressions, every odd prime greater than 10[sup]1999[/sup] + 100 can be expressed as a number of that form.

The fix? Change that to:

19. [B]General arithmetic progressions[/B], k * b[sup]n[/sup] + c, where c is a prime > 10[sup]2[/sup], where the prime is at least 2000 digits in length, and where the exponent n > 1

19 follows the same constraints as Proths as well, k < b[sup]n[/sup]

Now that 19 is fixed.

[B]Also: By #8, CRG assumes I made no restriction, also assumed they were something I thought up of. By, "Generalized Cullen/Woodall", I meant, [URL="http://primes.utm.edu/top20/page.php?id=42"]Generalized Cullen[/URL], or [URL="http://primes.utm.edu/top20/page.php?id=45"]Generalized Woodall[/URL].[/B]

Therefore, items 8, 9, 10, and 11 require no fix.

The objections to the k-b-b list that I think are coming up:

1. Every odd prime can be listed as k-b-b because (p-1) * (1[sup]1[/sup]) + 1 = p.
Solution: b > 1.

Following that is probably:
2. Every 4n + 1 number can be expressed as a k-b-b:
Solution: k < b[sup]b[/sup]

Woots! 905-digit General Cofactor:

[code]3209097023552191368355677463890027571749202779153366136089549940941127127165007377329013386450756151539054560263458120222230690308125353575523018705040704995781928789028289385995815544631936702436036685991696210190449953162883536653559552962280789764064066320035883153598322315965856759048928684221303117234955739737516760907718812535595462982047259256213277734351462879086885327984191355032588938735453708947253357907635258195017894339523292944404266913078107998717716397075013455490881287116136262864971948987162936201532412099553779176182193629232122172501581400104092790717875140544093678015378589891544975359389117315125749197635333868006485581362330284883221609216388775355643197427930406960545041935132118683264153128763681810822435576724781777826571304617063142534385633999934182327485225406637493027643273806648964785717210108084165988967744936735013024825136638609275468640807269967330554137517[/code]

Is the 905-digit cofactor of 170431718464782327252263897391988651686257088683807301979173668897417988033604374506998825918479837760556205541175012306612734123368718839 * p(180)[sup]2[/sup] + 1.

The other factor is a p137 factor, 49866720841082056288200444156902190141035815590392316055348443500498338562880272788480816736007987065964634953049533790956458476724195153

1089-digit Special Cofactor:

[code]701591878218588666347096831814973267440629265446182654861722158859690877626738114622546038069462125441285813793564786853876040461417279801797064618235492656477086618210880572353254905922359402998053449531537726006337550213987295790045395400647384874883162968814160232068771743425828444839580932713315261352095238764672198582042036473625585221768528459744204858591354843492815640774763773902888220487284694627978937116931333175659335178805365858441041098813676083651106629228060657463158077857675699267051832335471993741910284195175940185953289759659566954755910035993855524708608576636701751626692817355744841638072177503363079280931336985077823152654484562479733474207254824464368813067010536218621759992575503453833917134653022833767997582536934765509786671494526834916721186279891691526778471467343449981150101430033339136596563642969266992138548151484146690803846323268718136760202536696446029820267088785449879873610236505371606486072088918346795521412879944592013741721631684813288406486232391691555441381728587887074895184115122459339929328848606680058494012699716831688272059877601[/code]

The other 4 factors were:

[code]53517717056058200198206376035499716536692971012402956668067486105600000000000001 * 64064368137138073527168870335655198134314875723316242916986322944000000000000001 * 97621894304210397755685897654331730490384572530767608254455349248000000000000001 * 6972992450300742696834706975309409320741755180769114875318239232000000000000001[/code]


Okay: This is way too easy:

For Special and General cofactor:

Factorwork must be done and the user cannot know one or more of the factors beforehand. The smallest factor can be no smaller than 7 digits.

@CRG: That disables you from using anything such as "nextprime(10^6)" as a factor.

Now let's see how easy it is to find a large cofactor.

[QUOTE=3.14159;227414]Settling this quickly:

The first number, you typed, the last two, randomly-generated.[/QUOTE]

Ah, but this is false (even assuming that "randomly-generated" means "generated randomly or pseudorandomly"). However, it is possible that the numbers are somehow 'good enough' for you, which is why I asked the other three questions: how do you decide your answers for #1, #2, and #3?

[QUOTE=3.14159;227417]You can tear your heart out with as many complaints as you'd like, but that list is going to remain the way it is.[/QUOTE]

Fine by me -- just don't be annoyed if sm, someone else, or I post a record for those fields.

[QUOTE=3.14159;227417][B]Also: By #8, CRG assumes I made no restriction, also assumed they were something I thought up of. By, "Generalized Cullen/Woodall", I meant, [URL="http://primes.utm.edu/top20/page.php?id=42"]Generalized Cullen[/URL], or [URL="http://primes.utm.edu/top20/page.php?id=45"]Generalized Woodall[/URL].[/B][/QUOTE]

You make funny assumptions about what I assume.

[QUOTE=CRGreathouse]Fine by me -- just don't be annoyed if sm, someone else, or I post a record for those fields.
[/QUOTE]

I already fixed your whining complaints. And, why would I be annoyed? All decent-sized primes of those 20 types are welcomed.

Here is the list with the updates:

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 primorial 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]Special Cofactor[/B], where the prime cofactor is of one of the forms used in this list.
18. [B]General Cofactor[/B], where the prime cofactor is not of a special form.
19. [B]General arithmetic progressions[/B], k * b^n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length, and where the exponent n > 1.
20. [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.

[B]NOTE:[/B]

The same restrictions that apply to the Proth numbers apply to Factorial/Primorial/Generalized/Prime-based proths. For items 9, 10, and 11, the same restrictions that apply to Generalized apply to these.

For items 17 and 18, the factors cannot be known beforehand, and it must be one of the forms of prime in the list for Special Cofactor.

For item 19, the same restriction that is found in the Proths also applies, but c must still be greater than 100.

For item 20: Proth.exe or the original PrimeForm must be used without assistance to prove or disprove the primality of a PRP number. If you'd like, it can also be a completely untested candidate.

For items 12, 13, 14, and 15: The restriction of k < b[sup]b[/sup] applies as well.

For items 6 and 7: p(n) and n! must have an exponent of 1.

For items 1, 2, 3, and 4, the exponent must also be greater than 1.

For items 12, 13, 14, and 15, b must be greater than 1.

For item 2: The integer b must not be a primorial, factorial, or prime. This applies to Generalized Cullen/Woodall, item 8, and applies to item 12.

For item 5: Odd prime bases only. If you wish to use b = 2, go for item 1.

[QUOTE=CRGreathouse]You make funny assumptions about what I assume.
[/QUOTE]

You said you found a trivial weakness for those types of primes: Please, go ahead and post it.

yeah well if i was stupid I'd post 2^43112608*2^1+1 but you know this one already lol.

By decent-sized: The usual 1k-digit minimum, please. (750 for General Cofactor, and 1250 for Special Cofactor.)

[QUOTE=science_man_88]yeah well if i was stupid I'd post 2^43112608*2^1+1 but you know this one already lol.
[/QUOTE]

Mersenne numbers belong in General Cofactor, if you can prove that it is the cofactor of a number of the forms listed, [B]following the new rules[/B] (Bold = My emphasis.)

Also: For verification: Post the factoring session data for General/Special cofactor.

Also: If there have been distributed searches on k-b-b, can anyone point me to one, and what ranges were being searched? (This is to ensure I am not searching for what has already been searched for, like when I was searching for Generalized Fermats. Because they are already searched for up to 131072, up to 3 million, there is no point searching for them.)

I'm going to check to see the search ranges for Prime Sierpinski, since they search for items 2 or 5 (Base 5 Sierpinski project, if I remember correctly.)

The k * 4549[sup]4549[/sup] range I am testing is rather barren.

k = 180k to 360k.

Nothing so far up to 270k.

Update: Nevermind: 265134 * 4549[sup]4549[/sup] + 1 is a PRP. Off to prove it prime. (Does Proth's theorem also apply to Generalized Proth numbers? Answered my own question: Not to odd-number bases)

[code]Start: For n=4549 to 4549, For k=265134 to 265134 step 2, k*4549^n+1.
265134*4549^4549 + 1 may be prime. (a = 2)
265134*4549^4549 + 1 is prime! (verification : a = 3) [16646 digits][/code]

[QUOTE=3.14159;227427]Factorwork must be done and the user cannot know one or more of the factors beforehand. The smallest factor can be no smaller than 7 digits.[/QUOTE]

How can you verify this? Factoring data can presumably be reconstructed if you know the factors.

[QUOTE=3.14159;227438]I already fixed your whining complaints.[/QUOTE]

I'd have to check, but I'd guess you fixed less than 25% of them. Of course this is not your fault -- I was intentionally not explicit about most of them. I don't care to be: my purpose is not to tell you what to do but to cause you to think more deeply.

[QUOTE=3.14159;227438]And, why would I be annoyed? All decent-sized primes of those 20 types are welcomed.[/QUOTE]

Only for the same reason you seemed to not like #969.

[QUOTE=3.14159;227442]Mersenne numbers belong in General Cofactor, if you can prove that it is the cofactor of a number of the forms listed, [B]following the new rules[/B] (Bold = My emphasis.)

Also: For verification: Post the factoring session data for General/Special cofactor.[/QUOTE]

yeah I accidentally put +1 doh anyways if you keep changing the rules who cares not me.

[QUOTE=CRGreathouse]How can you verify this? Factoring data can presumably be reconstructed if you know the factors.
[/QUOTE]

You would have to go a long way to fake factor data. Meh, screw it. Too many people would take the time to fake everything.

To make sure it can't be faked: Smallest factor must at least be 65 digits and unknown to the user.

Yes, I'm sure someone would withhold it for 7-10 days and post it with fake data afterwards. Makes perfect sense.

Even better: Get rid of cofactor and replace it with complete factorization of a number of those forms.

[QUOTE=3.14159;227449]You would have to go a long way to fake factor data.[/QUOTE]

Maybe, maybe not. Perhaps I already have a program that fakes factor data.

Here's a Pari program that fakes trial division data
[code]fake(p,q)={

};[/code]

That was easy. Faking rho data also seems doable. NFS seems hard to fake, I'll admit, but just because I don't know of a way doesn't mean it's impossible.

But my point, like my point about your obsolete category, is that this suggests that the category itself is a bad idea, not that this small group of people would be likely to deceive each other.

Update: List is back down to 19 items. Restrictions still apply to the other items. I have gotten rid of cofactor and replaced it with completely factoring a number of the forms listed there. The smallest the number can be is 90 digits.

[QUOTE=3.14159;227449]To make sure it can't be faked: Smallest factor must at least be 65 digits and unknown to the user.[/QUOTE]

How can you verify that it was unknown to the user?

[QUOTE=3.14159;227449]Even better: Get rid of cofactor and replace it with complete factorization of a number of those forms.[/QUOTE]

That's fine -- but I still think we need a good way to judge the (prior) difficulty of finding a factorization.

[QUOTE=CRGreathouse]That's fine -- but I still think we need a good way to judge the (prior) difficulty of finding a factorization.
[/QUOTE]

See the above. Must be at least a c90, and it cannot be divisible by any prime smaller than 10 digits, and factors cannot be known to the user.

[QUOTE=3.14159;227453]See the above. Must be at least a c90.[/QUOTE]

So how do you rank two different factorizations? Say, a p30 . p70 vs. a p25 . p80? What about a p31 . p64? What about a p20 . p81? What about a p9 . p30 . p65?

[QUOTE=3.14159;227453]factors cannot be known to the user.[/QUOTE]

How do you determine this? Polygraph?

[QUOTE=CRGreathouse]So how do you rank two different factorizations? Say, a p30 . p70 vs. a p25 . p80? What about a p31 . p64? What about a p20 . p81? What about a p9 . p30 . p65?
[/QUOTE]

Simple: Semiprimes are most impressive, and it is also more impressive when the primes are evenly sized, differing in size by about 4-10 digits. By the way: Last one is invalid, smallest factor must be p10.

[QUOTE=CRGreathouse]How do you determine this? Polygraph?[/QUOTE]

Again, if you're willing to fake ECM/SIQS/NFS data, show me how.

I've tried to simplify the list in post #1011:

[code]
1. k*2^n+-1 Proth/Riesel
2. k*b^n+-1 Generalized Proth/Riesel
3. k*b!^n+-1 Factorial-based Proth/Riesel
4. k*b#^n+-1 Primorial-based Proth/Riesel
5. k*b^n+-1 Prime-based Proth/Riesel, b>2 prime
6. k*n#+-1 Primorial
7. k*n!+-1 Factorial
8. n*b^n+-1 Generalized Cullen/Woodall
9. n*b^n+-1 Factorial Cullen/Woodall, b Factorial
10. n*b^n+-1 Primorial Cullen/Woodall, b Primorial
11. n*b^n+-1 Prime-based Cullen/Woodall, b prime
12. k*b^b+-1 k-b-b, k<b^b
13. k*b^b+-1 k-b-b, b Factorial, k<b^b
14. k*b^b+-1 k-b-b, b Primorial, k<b^b
15. k*b^b+-1 k-b-b, b prime, k<b^b
16. 4-group n^1+1, n^2+1, n^4+1 primes
17. Factorization of form #1-#16, #digits>=90
18. Factorization not of form #1-#16, #digits>=90
19. k*b^n+c General arithmetic progressions, c>100 prime; #digits>=2000
20. k*b^n-1, k*b^n+1 Twins.

NOTE:
- n>1, b>1
[/code]

3.14159 searches only for +1-side on #1-#19.

[QUOTE=Karsten]Until no others changes made so far. Check it.
[/QUOTE]

Replace Special and General Cofactor with Factorization, also add that the smallest prime factor must be a p10 and the number must at minimum be a c90.

Why are there no k*b^n-1?

[QUOTE=Mini-Geek;227461]Why are there no k*b^n-1?[/QUOTE]

Because it's his own list! :smile:

[quote=Mini-Geek]Why are there no k*b^n-1?[/quote]

I knew one of you was going to complain about the -1 thing.

Other members only:

20. n-1 analogues of Proths.
21. n-1 analogues of k-b-b, (k * b[sup]b[/sup] - 1)
22. Twins.

I stay within 1-19. Ban me for a week if I defy that rule.

[QUOTE=3.14159;227456]Simple: Semiprimes are most impressive, and it is also more impressive when the primes are evenly sized, differing in size by about 4-10 digits. By the way: Last one is invalid, smallest factor must be p10.[/QUOTE]

"Semiprimes are most impressive": Does this mean that a p20 . p70 is better than a p80 . p80 . p80?

[QUOTE=3.14159;227456]By the way: Last one is invalid, smallest factor must be p10.[/QUOTE]

Yes, that was intentional. I'm amused that a p20 . p70 qualifies while a p9 . p90 . p90 does not.

As opposed to 265134* 4549[sup]4549[/sup], I found a PRP for b = 4861 pretty soon:

240790 * 4861[sup]4861[/sup] + 1 (17927 digits)

[QUOTE=CRGreathouse]"Semiprimes are most impressive": Does this mean that a p20 . p70 is better than a p80 . p80 . p80?
[/QUOTE]

Okay: Large composites with prime factors no smaller than 50 digits are more impressive.

[QUOTE=CRGreathouse]Yes, that was intentional. I'm amused that a p20 . p70 qualifies while a p9 . p90 . p90 does not.
[/QUOTE]

Well, factorization cannot have any restrictions then, except the number must be at least a c90. Factor restriction lifted.

Now, to look for a relatively unsearched prime: Primorial. I'll use 45361.

[QUOTE=3.14159;227467]Well, factorization cannot have any restrictions then, except the number must be at least a c90. Factor restriction lifted.[/QUOTE]

Sounds good.

[QUOTE=3.14159;227467]Okay: Large composites with prime factors no smaller than 50 digits are more impressive.[/QUOTE]

I was hoping for a general way to transform a vector of the sizes of the prime factors (and possibly a composite cofactor?) into a score that could be ranked, so if I had 100 such factorizations I could decide what was the best, the second best, and so forth. At the origin of your list you used (or seemed to use) "points = size of largest prime factor", which I didn't like because it made a p55 . p90 sound better than a p88 . p89.

[QUOTE=CRGreathouse]Sounds good.
[/QUOTE]

Excellent.


[QUOTE=CRGreathouse]I was hoping for a general way to transform a vector of the sizes of the prime factors (and possibly a composite cofactor?) into a score that could be ranked, so if I had 100 such factorizations I could decide what was the best, the second best, and so forth. At the origin of your list you used (or seemed to use) "points = size of largest prime factor", which I didn't like because it made a p55 . p90 sound better than a p88 . p89.
[/QUOTE]

What makes it better is how hard it was to factor it.

Also: Factorization = [B]Complete[/B] factorization, unless it is too large.

[QUOTE=3.14159;227470]What makes it better is how hard it was to factor it.[/QUOTE]

So, in short, you don't have a method for determining the 'winners' for #17 and #18, since you have no way of deciding which of two factorizations is more difficult.

Not a problem for me, but I imagine that would discourage people from submitting an entry.

[QUOTE=CRGreathouse]So, in short, you don't have a method for determining the 'winners' for #17 and #18, since you have no way of deciding which of two factorizations is more difficult.
[/QUOTE]

Well, it depends on the methods you use to factor the number. It would be damn impressive if you factored a number into p60 * p75 by using trial-division alone.

[QUOTE=3.14159;227472]Well, it depends on the methods you use to factor the number. It would be damn impressive if you factored a number into p60 * p75 by using trial-division alone.[/QUOTE]

A person could make that claim, of course. Or a person could use a nondeterministic method (pick a factor and test).

But this doesn't solve the problem of choosing a winner. Are you going to have different categories for different methods? (And what counts as a different method?) And within a given method, how do you decide what is better?

[QUOTE=CRGreathouse]A person could make that claim, of course. Or a person could use a nondeterministic method (pick a factor and test).
[/QUOTE]

But, how many times would they need to do so before guessing the correct factor?

[QUOTE=CRGreathouse]But this doesn't solve the problem of choosing a winner. Are you going to have different categories for different methods? (And what counts as a different method?) And within a given method, how do you decide what is better?
[/QUOTE]

For individual factoring methods? And, concerning what is most impressive, that should be obvious. Numbers with large factors, relative to their size.

At the moment, I'm searching for k * 45361# + 1 (≈ 19605-19610 digits).

I should get something by 01:00.

[QUOTE=3.14159;227478]But, how many times would they need to do so before guessing the correct factor?[/QUOTE]

Just once, if they're lucky enough.

I'm trying to help you determine reasonable methods for judging your lists and you seem determined to shrug it off. Ah well.


[QUOTE=3.14159;227478]For individual factoring methods? And, concerning what is most impressive, that should be obvious. Numbers with large factors, relative to their size.[/QUOTE]

Not obvious at all, because you have multiple factors. If two factorizations have different numbers of factors, or if they have the same number of factors and (sorting them by size) there are corresponding factors that are smaller for one but also a pair of corresponding factors where the other is larger, how do they compare?

p30 . p90 vs. p35 . p85 vs. p34 . p 89.

If you don't have a way to rank them, you don't have a way to decide winners for your two cofactor 'competitions'.

[quote=CRGreathouse]I'm trying to help you determine reasonable methods for judging your lists and you seem determined to shrug it off. Ah well.[/quote]

All the items are fine, except for Factorization.

[quote=CRGreathouse]If you don't have a way to rank them, you don't have a way to decide winners for your two cofactor 'competitions'.[/quote]

Size of the smallest factor is what places a number as high or low in the scale.

Ex: A number with a smallest factor of 6874713856829275576652590301803216341934585163 is a higher-ranking number than a number with a smallest factor of 86275185784708979, and a number with a smallest factor of 7062362420427661148487418730654866333839957380354935428654077083643909 is higher-ranking than that which has a smallest factor of 6874713856829275576652590301803216341934585163.

Updates for Special Cofactor and General Cofactor (I decided to keep them):

The size of the [B]largest[/B] prime factor is what determines how impressive it is:

Yes, CRG, even 3 * p10000 will be accepted into either Special or General cofactor. The hard part will be proving that the 10000-digit number is a prime number. ECPP is recommended.

It must also conform to restrictions placed in post 1011, for Special Cofactor.

For General Cofactor (Also shared with Special Cofactor), the number must be at least 1000 digits in length.

And, finally:

[B]Reminder:[/B] I, (optionally mods), search for the + 1 primes for items 1-19 and twin primes.

Other members, (optionally mods), look for the - 1 primes for items 1-19 and twin primes.

Next notice:

For Special and General Cofactor:

For + 1 searchers, which are me alone, and optionally, the mods:

The cofactor, for Special cofactor must be a + 1 number!

Same goes for - 1 searchers: All other members, mods optional:

Cofactor must be a -1 number!

[QUOTE=3.14159;227483]The size of the [B]largest[/B] prime factor is what determines how impressive it is:

Yes, CRG, even 3 * p10000 will be accepted into either Special or General cofactor.[/QUOTE]

10^10000 + 336030 = p1 . p1 . p10000, see [url]http://oeis.org/classic/A142587[/url]

[QUOTE=CRGreathouse]10^10000 + 336030 = p1 . p1 . p10000, see [url]http://oeis.org/classic/A142587[/url][/QUOTE]

You go a long way to try and use your methods.

[QUOTE=3.14159]You go a long way to try and use your methods.
[/QUOTE]

@CRG: Now, I will truly congratulate you if you can give me a p(10^6) cofactor.

Oh, right: 3 * (2[sup]43112609[/sup]-1)?

[QUOTE=3.14159;227486]@CRG: Now, I will truly congratulate you if you can give me a p(10^6) cofactor.

Oh, right: 3 * (2[sup]43112609[/sup]-1)?[/QUOTE]

Clearly, you have no appreciation for my minimalism. I would choose
3481 · 2[SUP]6816832[/SUP] - 59 · 2[SUP]3408417[/SUP] + 1.

[QUOTE]Clearly, you have no appreciation for my minimalism. I would choose
59[sup]2[/sup] · 2[sup]6816832[/sup] - 59 · 2[sup]3408417[/sup] + 1.[/QUOTE]

You tend to choose simple expressions. Why the sudden change of heart?

Also: I'm sieving for k * 15661[sup]15661[/sup] + 1

P.S: Dammit, guys. You could have helped out with the searches.

Sieved to 1.63 * 10[sup]12[/sup] for k * 13751[sup]13751[/sup] + 1

[QUOTE=3.14159;227489]You tend to choose simple expressions. Why the sudden change of heart?[/QUOTE]

Bigger number, smaller cofactor (while still being at least a million digits).

Okay: For now, I'm going to be sieving. I have finally convinced myself to look for larger primes, optimally in the 30-70k digit range.

I have a 56300 and 65700 prime-based k-b-b search.

Okay; Mods? No assistance? Come on, toss me a bone here!

Well, in the meantime... A General Cofactor is coming up.

[QUOTE=3.14159;227492]Okay; Mods? No assistance? Come on, toss me a bone here![/QUOTE]

Exactly what kind of assistance are you looking for from the Mods?

[QUOTE=axn]Exactly what kind of assistance are you looking for from the Mods?
[/QUOTE]

Some assistance with the searches.

By mods, I mean, the mods that visit here the most often. (Karsten, etc.)

Also from other regular members.

[QUOTE=3.14159;227495]By mods, I mean, the mods that visit here the most often. (Karsten, etc.)[/QUOTE]

I've got enough open work to do and will not waste my CPU-time for the search of lonesome 70k-digit primes unless they'll close some holes or new n-ranges in my [url=www.rieselprime.de]Riesel DataBase[/url]!

yeah I may only do factoring as what I was given as dates says lucas lehmer test for one exponent wouldn't finish in almost 4 years.

[QUOTE=Karsten]I've got enough open work to do and will not waste my CPU-time for the search of lonesome 70k-digit primes unless they'll close some holes or new n-ranges in my Riesel DataBase!
[/QUOTE]

Could have just said, "No, thanks, I'm busy."

This was unnecessarily rude.

Also: Since you're not willing to aid in the search for smaller prime numbers: Would you be willing to aid in the search for a 257920-digit prime?

No, thanks, I'm busy.

[QUOTE=Karsten]No, thanks, I'm busy.
[/QUOTE]

What range are you working on at the moment?

Sieved up to 1.467 * 10[sup]12[/sup] for k * 15661[sup]15661[/sup] + 1. Approximately 1 in 25 candidates are remaining.

Most of what I am going to do is sievework, for now.

The next three searches are going to be prime k-b-b searches, or item 15.

Maybe I will look for item 16: Number, square, and fourth.

The smallest example = 2:

3, 5, 17.

Followed by 4: 17, 257, 65537.

Sample search in PARI:

[code]64951 and 4218502501 and 17795763342506250001 are primes
68041 and 4629441601 and 21431729527810560001 are primes
78691 and 6192116101 and 38342301795879210001 are primes
78901 and 6225210001 and 38753239544100000001 are primes
79537 and 6325975297 and 40017963445602287617 are primes
80287 and 6445841797 and 41548876459060505617 are primes
80677 and 6508616977 and 42362094940275384577 are primes
83047 and 6896638117 and 47563617303064029457 are primes
83617 and 6991635457 and 48882966349596327937 are primes
85147 and 7249841317 and 52560199107180611857 are primes
86077 and 7409077777 and 54894433490817106177 are primes
91807 and 8428341637 and 71036942733131156497 are primes
92221 and 8504528401 and 72327003306406560001 are primes
92467 and 8549961157 and 73101835769108856337 are primes
92641 and 8582169601 and 73653635043164160001 are primes
92767 and 8605530757 and 74055159592461931537 are primes
95971 and 9210240901 and 84828537436032810001 are primes
96181 and 9250592401 and 85573459750937760001 are primes
100927 and 10186057477 and 103755766904375490577 are primes
101917 and 10386871057 and 107887090333970555137 are primes
103657 and 10744566337 and 115445705748704464897 are primes
105997 and 11235152017 and 126228640822628864257 are primes
106747 and 11394708517 and 129839382164602922257 are primes
107251 and 11502562501 and 132308944066406250001 are primes
109267 and 11939058757 and 142541123979220267537 are primes
112087 and 12563271397 and 157835788169551788817 are primes
114661 and 13146915601 and 172841389793523360001 are primes[/code]

[QUOTE=3.14159;227509]Maybe I will look for item 16: Number, square, and fourth.[/QUOTE]

This is the one that amuses me the most. You could use a nice sieve for this form.

[QUOTE=CRGreathouse]This is the one that amuses me the most. You could use a nice sieve for this form.
[/QUOTE]

I wonder how that would work...

Oh. Right. kmin, kmax, multiplier(n), pmax.

What's your record for that form, anyway?

[QUOTE=3.14159;227513]I wonder how that would work...

Oh. Right. kmin, kmax, multiplier(n), pmax.[/QUOTE]

nmin, nmax, and pmax I'd think. You only have one variable in the expression... right?

[QUOTE=CRGreathouse]What's your record for that form, anyway?
[/QUOTE]

Number, square, and fourth? 211 digits, for the largest number, the fourth power.

[QUOTE=CRGreathouse]nmin, nmax, and pmax I'd think. You only have one variable in the expression... right?
[/QUOTE]

I normally choose the multiplier and k-range.