PARI's commands

[3.14159]

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

[CRGreathouse]

Originally Posted by 3.14159

Nothing to respond to in Post 993.

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

[3.14159]

Originally Posted by CRGreathouse

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

Settling this quickly:

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

[science_man_88]

Originally Posted by 3.14159

Updated list:
1. Proths, where b is 2.
2. Generalized Proths, where b is any integer that is not a factorial, primorial, or prime number.
3. Factorial-based proths, where b is a factorial number.
4. Primorial-based proths, where b is a primorial number.
5. Prime-based proths, where b is a prime number.
6. Primorial, k * p(n) + 1
7. Factorial, k * n! + 1
8. Generalized Cullen/Woodall, k * b^k + 1, where b is any integer that is not a factorial, primorial, or prime number.
9. Factorial Cullen/Woodall, where b, optionally k, is a factorial number.
10. Primorial Cullen/Woodall, where b, optionally k, is a primorial number.
11. Prime-based Cullen/Woodall, where b is a prime number
12. k-b-b, numbers of the form k * b^b + 1, where b is any integer that is not a factorial, primorial, or prime number.
13. Factorial k-b-b, where b, optionally k, is a factorial number.
14. Primorial k-b-b, where b, optionally k, is a primorial number.
15. Prime-based k-b-b, where b is a prime number.
16. Number, square, and fourth, where n^1 + 1, n^2 + 1, and n^4 + 1 are all primes.
17. Special Cofactor, where the prime cofactor is of one of the forms used in this list.
18. General Cofactor, where the prime cofactor is not of a special form.
19. General arithmetic progressions, k * n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length.
20. Obsolete-tech-proven primes, 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.

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

[3.14159]

Originally Posted by 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

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 10¹⁹⁹⁹ + 100 can be expressed as a number of that form.

The fix? Change that to:

19. General arithmetic progressions, k * bⁿ + c, where c is a prime > 10², 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 < bⁿ

Now that 19 is fixed.

Also: By #8, CRG assumes I made no restriction, also assumed they were something I thought up of. By, "Generalized Cullen/Woodall", I meant, Generalized Cullen, or Generalized Woodall.

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) * (1¹) + 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^b

[3.14159]

Woots! 905-digit General Cofactor:

3209097023552191368355677463890027571749202779153366136089549940941127127165007377329013386450756151539054560263458120222230690308125353575523018705040704995781928789028289385995815544631936702436036685991696210190449953162883536653559552962280789764064066320035883153598322315965856759048928684221303117234955739737516760907718812535595462982047259256213277734351462879086885327984191355032588938735453708947253357907635258195017894339523292944404266913078107998717716397075013455490881287116136262864971948987162936201532412099553779176182193629232122172501581400104092790717875140544093678015378589891544975359389117315125749197635333868006485581362330284883221609216388775355643197427930406960545041935132118683264153128763681810822435576724781777826571304617063142534385633999934182327485225406637493027643273806648964785717210108084165988967744936735013024825136638609275468640807269967330554137517

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

The other factor is a p137 factor, 49866720841082056288200444156902190141035815590392316055348443500498338562880272788480816736007987065964634953049533790956458476724195153

[3.14159]

1089-digit Special Cofactor:

701591878218588666347096831814973267440629265446182654861722158859690877626738114622546038069462125441285813793564786853876040461417279801797064618235492656477086618210880572353254905922359402998053449531537726006337550213987295790045395400647384874883162968814160232068771743425828444839580932713315261352095238764672198582042036473625585221768528459744204858591354843492815640774763773902888220487284694627978937116931333175659335178805365858441041098813676083651106629228060657463158077857675699267051832335471993741910284195175940185953289759659566954755910035993855524708608576636701751626692817355744841638072177503363079280931336985077823152654484562479733474207254824464368813067010536218621759992575503453833917134653022833767997582536934765509786671494526834916721186279891691526778471467343449981150101430033339136596563642969266992138548151484146690803846323268718136760202536696446029820267088785449879873610236505371606486072088918346795521412879944592013741721631684813288406486232391691555441381728587887074895184115122459339929328848606680058494012699716831688272059877601

The other 4 factors were:

53517717056058200198206376035499716536692971012402956668067486105600000000000001 * 64064368137138073527168870335655198134314875723316242916986322944000000000000001 * 97621894304210397755685897654331730490384572530767608254455349248000000000000001 * 6972992450300742696834706975309409320741755180769114875318239232000000000000001

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.

[CRGreathouse]

Originally Posted by 3.14159

Settling this quickly:

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

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?

[CRGreathouse]

Originally Posted by 3.14159

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.

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

Originally Posted by 3.14159

Also: By #8, CRG assumes I made no restriction, also assumed they were something I thought up of. By, "Generalized Cullen/Woodall", I meant, Generalized Cullen, or Generalized Woodall.

You make funny assumptions about what I assume.

[3.14159]

Originally Posted by CRGreathouse

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

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. Proths, where b is 2.
2. Generalized Proths, where b is any integer.
3. Factorial-based proths, where b is a factorial number.
4. Primorial-based proths, where b is a primorial number.
5. Prime-based proths, where b is a prime number.
6. Primorial, k * p(n) + 1
7. Factorial, k * n! + 1
8. Generalized Cullen/Woodall, k * b^k + 1
9. Factorial Cullen/Woodall, where b, optionally k, is a factorial number.
10. Primorial Cullen/Woodall, where b, optionally k, is a primorial number.
11. Prime-based Cullen/Woodall, where b is a prime number
12. k-b-b, numbers of the form k * b^b + 1
13. Factorial k-b-b, where b, optionally k, is a factorial number.
14. Primorial k-b-b, where b, optionally k, is a primorial number.
15. Prime-based k-b-b, where b is a prime number.
16. Number, square, and fourth, where n^1 + 1, n^2 + 1, and n^4 + 1 are all primes.
17. Special Cofactor, where the prime cofactor is of one of the forms used in this list.
18. General Cofactor, where the prime cofactor is not of a special form.
19. General arithmetic progressions, 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. Obsolete-tech-proven primes, 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.

NOTE:

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^b 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.

Originally Posted by CRGreathouse

You make funny assumptions about what I assume.

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

[science_man_88]

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