Take any number and keep appending 1's to the right of it. Are there an infinite number of primes in this sequence? Ignoring sequences that are always factorable such as starting with 11, Can we take any other number such as 42 and continually append 1s (forming the sequence {42, 421, 4211, ...}) to get a sequence that has an infinite number of primes in it?
 A: Unless prevented by congruence restrictions, a sequence that grows exponentially, such as Mersenne primes or repunits or this variant on repunits, is predicted to have about $c \log(n)$ primes among its first $n$ terms according to "probability" arguments.  Proving this prediction for any particular sequence is usually an unsolved problem. 
There is more literature (and more algebraic structure) available for the Mersenne case but the principle is the same for other sequences.
http://primes.utm.edu/mersenne/heuristic.html
Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The New Mersenne Conjecture." Amer. Math. Monthly 96, 125-128, 1989
A: I think this is an open question. Lenny Jones gave a talk in which he noted that the numbers 12, 121, 1211, 12111, 121111, etc., are all composite - until you get to the one with 138 digits, that's a prime. 
Jones' work appears in the paper, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite integers?, Amer. Math Monthly 118 (Feb. 2011) 153-160. He finds that 37 is the smallest positive integer such that you get nothing but composites by appending any positive number of ones. It seems to be easier to find a sequence with no primes than a sequence which you can prove has infinitely many. 
A: So it can't be true or can't be even false. As we know that primes gaps increases when we see large prime numbers. It is true there are infinitely many primes but these gaps really create the trouble to verified whether these types of prime numbers may exist. Let take an example let n=3 so the sequence will run like 
(31, 311, 3111, 31111,...) in this sequence every third term will divisible by 3 and if we go higher the prime gaps increases but as we know the sequence is infinite and following a pattern then it can hit some prime numbers also, by this assumption we can state yes prime numbers like do exist till infinity. 
In my knowledge I don't know any mathematical way to proof it except this above hypothesis.
