Strings of Digits
 

Where is the longest string of consecutive 0's in a known prime?
What about 1's, 2's, 3's, etc.?
What if we restrict to the known Mersenne primes?

[QUOTE=davar55;147486]Where is the longest string of consecutive 0's in a known prime?[/QUOTE][pedantic]
Can we assume you are not referring to leading zeros? Can we assume you are not referring to trailing zeros after the radix point? Can we assume base 10?
[/pedantic]

[QUOTE=davar55;147486]Where is the longest string of consecutive 0's in a known prime?
What about 1's, 2's, 3's, etc.?
What if we restrict to the known Mersenne primes?[/QUOTE]

Using :[URL="http://primes.utm.edu/primes/search.php"]http://primes.utm.edu/primes/search.php[/URL]
and searching for 10^ we can get probably the best known solutions for digit=0 and 9: p=98035*10^196070+1 containing 196069 consecutive 0's.
q=99*10^139670-1 containing 139670 consecutive 9's. Since both of them are on the top5000 list, it is possible to check remaining primes on the list, if there is a better solution or not.

[quote=retina;147487][pedantic]
Can we assume you are not referring to leading zeros? Can we assume you are not referring to trailing zeros after the radix point? Can we assume base 10?
[/pedantic][/quote]

Hmm ... no lead zeros and only integral parts. But if we allow other bases,
all bets but 1111111........1111111 are off !

n is the largest number of consecutive digits d in the 46 known Mersenne primes 2^p-1 with index m=1..46.

d n m p
0 9 46 43112609
1 9 43 30402457
2 8 40 20996011
3 8 40 20996011
4 7 43 30402457
4 7 45 37156667
4 7 46 43112609
5 7 39 13466917
5 7 41 24036583
5 7 43 30402457
5 7 44 32582657
5 7 45 37156667
5 7 46 43112609 (3 times)
6 7 42 25964951
6 7 44 32582657
6 7 46 43112609
7 7 43 30402457
7 7 45 37156667 (2 times)
7 7 46 43112609
8 8 43 30402457
9 8 39 13466917

Fot other digits than 0 and 9, the best I could spot in the complete Prime Pages database is [URL="http://primes.utm.edu/primes/page.php?id=16887"](8*10^11083-8*10^1003+9)/9[/URL] which starts with 10080 8's. That would be easy to beat with a new prime.