The sequence a(n) = 12013n has a divergent reciprocal sum but no two terms are within 12012. Do you have any way of removing these trivial kinds of counterexamples? Charles Greathouse Analyst/Programmer Case Western Reserve University On Wed, Jun 26, 2013 at 8:23 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
Last month, Zhang proved that there exists a number N such that there are infinitely many primes that differ from another prime by not more than N. (He showed that N is at most 70 million. That upper bound has since been reduced to 12,012. See
http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_pri... )
I've wondered if the same is true for any monotonically increasing sequence of positive integers (i.e. no duplicate terms) for which the sum of the reciprocals diverges. Can anyone think of a counterexample?
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