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?

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.
