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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?

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In other words, does the sequence $a_n = 10^n m + (10^n-1)/9$, $n=1,2,3,\dots$, $m\in\mathbb{N}$ contain an infinite number of primes. This is from using the geometric sum formula. Using this, the question can be posed in arbitrary number base instead of just decimal. – anon Jul 13 '11 at 5:51
No reason to ignore stuff that begins with $11$. It is not known whether there are infinitely many primes of the shape $111 \dots 11$. It is the only problem in the family of problems you mention that has a fairly big literature. Numbers of shape $111\dots 11$ are called repunits. My feeling is that any of your questions, like the one about stuff that starts with $42$, is exceedingly difficult. – André Nicolas Jul 13 '11 at 6:13
@user6312 Oops, I somehow thought that all repunits were composite. How wrong I am. – Paul Jul 13 '11 at 6:24
up vote 14 down vote accepted

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.

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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.

Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The New Mersenne Conjecture." Amer. Math. Monthly 96, 125-128, 1989

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