Prime posting thread, part 2. (With a catch.)

3.14159 · 2010-10-29, 18:48

Again, post many, many primes.

But, this time:

1. At least 200 digits.
2. Composed of only two or three numbers.
3. PRPs accepted, although proof is recommended.

Let me kick things off..

Code:

99494999494944444449994494999444999999449499999449949944999949999994994944499944949949449494444994994994944449999999994999949449449999994494944994499994494499999994444949994999449494949944444999949949999499999999994494999499449949

is prime, and composed of only numbers 4 and 9.

And;

Code:

49999999999494949194499149994411991449199494194944991994199999444949141494994499949944919949199919119444999449194944991494144994999494994444149499499944991444494944444919194449949994949449491114499941444999119994919419449999919919494994194999419944114499999941199994494491499994911449149491919499149449119994994949914994991999944194994919144991419499444449994441141944919994999149114994914999419999944494499414449141941994499144991491499999194994119491199194194999914441499444999419494149444491949991419449449144494199444419994944944991994944491191494994144999919919999999949194149414949449449499949441491199499914449494919494999991194191944999994199

230 and 650 digits. The latter, only digits 1, 4, and 9.

Or, rather, are both PRP.

3.14159 · 2010-10-29, 18:57

Code:

6666677676677766767666667776767676666666677767767777776667776776666667767767666666767667767766776676677677777676767767776677776766777777767766767777677676677766666766767676667666677777766766776766676767777667777767777767676776766667777766777666676677

250-digit prime, using only 6 and 7.

Code:

9999995999995599999599999595959999999999959995959959959999999999599959995959599559999999999599999995559999999959995999999595955995999995995999999595559999999555999999999595559595959999959999999999559959999999999999595995559995999999959999999999999959

and also

Code:

599999959555559999595559555999995959955999955999995999995959955555599599995955599959999995999959959999999999955595999599995999995599599959999555959559955559959559995959999959959595599995999595555955959955999955995999999999995995995595595959995999995595999999555599995595999959995995999995995559959995995599955955955995959999999995995999959599999999959999959995555999595959995959599999959955999959995555959999999959595959999959595959555995559999999959

Batalov · 2010-10-29, 22:02

(10¹²⁸⁹¹+11)/3 has all 3s and a 7 in the end. 12891 digits; tricky to prove (if not for some luck...) - because of the denominator. Just so that it doesn't get lost in that other thread. This one is proven just recently.

A good list of such primes and PRPs is here and here -- it is somewhat important to know the existing records, right? A yardstick? (Like in the 'Battle of sexes' thread, in order to beat women's world record it would be a good idea to first find out how much is it exactly.)

CRGreathouse · 2010-10-30, 00:47

Well, in the usual spirit of my contributions, I give you

10¹⁹⁹ + 91119

which is the smallest possible entry.

The corresponding smallest with only two distinct decimal digits is

10¹⁹⁹ + 1110101011.

Both of these are, of course, P200s.

If I was to cheat and use binary, it would be 2¹⁹⁹ + 101 =

Code:

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100101₂

but this is only a P60.

3.14159 · 2010-10-31, 21:15

By the way, is there a prime of this type, such that no smaller substring of it is ever prime?

If so, which is the largest for which that's true?

Batalov · 2010-10-31, 21:41

Quote:

Originally Posted by 3.14159

By the way, is there a prime of this type, such that no smaller substring of it is ever prime?

If so, which is the largest for which that's true?

Good question (even though most probably a solved problem, but a good excercise to paper-and-pencil from scratch).

E.g. 999999999*10^180230-1 doesn't satisfy: substring 89 is prime.
Similarly a 4-plus-9ers will fail with 499 substrings...
1-plus-0ers stand a good chance, if they stay clear of 11 substrings.
Interesting...

TimSorbet · 2010-10-31, 21:59

10^616+1001001 and 10^668+1001001 match the criteria. 1001 and 1001001 are composite, as are the leading 100...001 and 100...001001 for each number. They're PRPs right now, but I'm proving them.
http://factordb.com/index.php?id=1100000000225031459 (Edit: proven prime by the DB)
http://factordb.com/index.php?id=1100000000225031635 (Edit 2: proven prime by me, proof attached; Edit 3: and by the DB)

Batalov · 2010-11-01, 00:21

10^n+k with k that spells three 1's and the rest 0's is a good ticket.
This is because substrings of it will have 0,1,2,3 or 4 1's (the last one is itself). However substrings with zero, one and three 1's are easily dismissed (the latter is divisible by 3). So all one needs for the number to belong in the class is to show compositeness of substrings with two 1's, which is not hard. When n is large, these will remain PRPs.

k*10^n+1 are even better because they will be proven primes!

science_man_88 · 2010-11-01, 00:32

Quote:

Originally Posted by 3.14159

By the way, is there a prime of this type, such that no smaller substring of it is ever prime?

If so, which is the largest for which that's true?

Code:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

lets see what's taken out:
any number or group that has any of the digits:2,3,5,7
any number or group that has any of the sets(2):1'1,1'4(if taken out of that order = 41),1'6(if taken out of that order = 61),1'9,8'9, etc. any way

science_man_88 · 2010-11-01, 00:34

Quote:

Originally Posted by Batalov

10^n+k with k that spells three 1's and the rest 0's is a good ticket.
This is because substrings of it will have 0,1,2,3 or 4 1's (the last one is itself). However substrings with zero, one and three 1's are easily dismissed (the latter is divisible by 3). So all one needs for the number to belong in the class is to show compositeness of substrings with two 1's, which is not hard. When n is large, these will remain PRPs.

k*10^n+1 are even better because they will be proven primes!

616 61 is prime so that's out last i checked lol.

CRGreathouse · 2010-11-01, 00:47

Quote:

Originally Posted by 3.14159

By the way, is there a prime of this type, such that no smaller substring of it is ever prime?

This is Sloane's A033274.

Quote:

Originally Posted by 3.14159

If so, which is the largest for which that's true?

It appears to be infinite. Zak Seidov finds 1000 the last of which is 9468169.

2010-10-30, 00:47	#4
CRGreathouse Aug 2006 5,987 Posts	Well, in the usual spirit of my contributions, I give you 10¹⁹⁹ + 91119 which is the smallest possible entry. The corresponding smallest with only two distinct decimal digits is 10¹⁹⁹ + 1110101011. Both of these are, of course, P200s. If I was to cheat and use binary, it would be 2¹⁹⁹ + 101 = Code: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100101₂ but this is only a P60. Last fiddled with by CRGreathouse on 2010-10-30 at 00:50

2010-10-31, 21:15	#5
3.14159 May 2010 Prime hunting commission. 2⁴·3·5·7 Posts	By the way, is there a prime of this type, such that no smaller substring of it is ever prime? If so, which is the largest for which that's true? Last fiddled with by 3.14159 on 2010-10-31 at 22:00

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2010-10-29, 22:02	#3
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 23443₈ Posts	(10¹²⁸⁹¹+11)/3 has all 3s and a 7 in the end. 12891 digits; tricky to prove (if not for some luck...) - because of the denominator. Just so that it doesn't get lost in that other thread. This one is proven just recently. A good list of such primes and PRPs is here and here -- it is somewhat important to know the existing records, right? A yardstick? (Like in the 'Battle of sexes' thread, in order to beat women's world record it would be a good idea to first find out how much is it exactly.)

2010-11-01, 00:21	#8
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 43×233 Posts	10^n+k with k that spells three 1's and the rest 0's is a good ticket. This is because substrings of it will have 0,1,2,3 or 4 1's (the last one is itself). However substrings with zero, one and three 1's are easily dismissed (the latter is divisible by 3). So all one needs for the number to belong in the class is to show compositeness of substrings with two 1's, which is not hard. When n is large, these will remain PRPs. k*10^n+1 are even better because they will be proven primes!