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Apr 9, 2012 at 19:12 comment added Noam D. Elkies Not only is it at most the number of coprime pairs in $[1,N]$ but indeed exactly that number. If $a/b \equiv c/d \bmod p$ then $ad-bc$ is a multiple of $p$, and $|ad-bc| \leq N^2 < p$ so $ad-bc=0$ and $a/b=c/d$.
Apr 9, 2012 at 17:24 vote accept Stanley Yao Xiao
Apr 9, 2012 at 17:19 history answered Seva CC BY-SA 3.0