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Consider the sets of consecutive positive integers:\

$A = \{ a,a+1,...,a+n-1 \}$,\

$B = \{b, b+1, ..., b + m - 1\}$.\

where $n$, $m \in \mathbb{Z}$ with $3 \leq n \leq m$. Is there a formula for a lower bound, in terms of $m$ and $n$, on the number of relatively prime pairs $(x,y)$ where $x \in A$ and $y \in B$?

In short, given two lists of consecutive positive integers, each with at least 3 terms, is there a lower bound on the number of relatively prime pairs where the first term of the pair comes from the first list and the second from the second list?

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  • $\begingroup$ In terms of what? $\endgroup$ – Jorge Fernández Hidalgo Nov 17 '15 at 20:45
  • $\begingroup$ In terms of $m$ and $n$, the length of the list. For example, let $n = 20$ and $m = 23$. Given any list of 20 consecutive integers and another of 23 consecutive integers, is there a lower bound on the number of relatively prime pairs, one term from each pair coming from each list, where the bound is given in terms of 20 and 23? Sorry if this is confusingly worded; I'm not the best at explaining. $\endgroup$ – user166293 Nov 17 '15 at 21:03

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