On 6/26/2013 8:23 PM, Keith F. Lynch 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?
Here's a counterexample. Take the sequence that consists of all the 1-digit numbers, every other 2-digit number, every third 3-digit number, etc. There's some freedom in choosing the first entry of each length; do it so that the sequence of first differences is non-decreasing. For example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ..., 96, 98, 100, 103, ..., 994, 997, 1000, 1004, ..., 9992, 9996, 10000, 10005, ..., 99990, 99995, 100000, 100006, ..., 999988, 999994, 1000000, 1000007, ..., 9999991, 9999998, 10000005, 10000013, ... (It took a while before 10^n wasn't in the arithmetic progression on the previous line!) The first differences grow, and the sum of the reciprocals diverges because we have (roughly) 0.9*10^n / n terms on the nth line, each less than 10^n, so their reciprocals add up to more than 0.9/n. -- Fred W. Helenius fredh@ix.netcom.com