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All Random Solved Random Open
OPEN
Let $a\geq 1$. Must there exist some $b>a$ such that \[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},\] with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?
For example, \[\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.\]