Finding positive integer triples $(m,n,r)$ with $\frac{mr}{m+n}$ being an integer and $r-n > \frac{mr}{m+n}-1$ and $r\le \max{(m,n)}$ [closed]

Question

I would like to use Mathematica to find all positive integer triples $(m,n,r)$ satisfying

(1) $\frac{mr}{m+n}$ is an integer;

(2) $r-n > \frac{mr}{m+n}-1$;

(3) $r\le \max{(m,n)}$;

(4) $1\le m < 1000$, $1\le n < 1000$, $1\le r < 1000$, all integers.

The code I use is:

   FindInstance[m*r ==(m+n)*s && r-n > m*r/(m+n)-1 && r <= Max[m,n] && 0<m<1000 && 0<n<1000 && 0<r<1000, {m,n,r,s}, Integers]

But I got an error that I don't understand

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Update: As pointed by azerbajdzan, I should use Max instead of FindMaximum (corrected above for convenience of future reference.) I will still happy to see anyone who could improve the code, making it run faster.

Answer 1

Since m*r/(m+n) == s, we replace r - n > m*r/(m + n) - 1 with r - n > s - 1

Reduce[{m*r == (m + n)*s, r - n > s - 1, 
  r <= Max[m, n], {m, n, r} > 0}, Integers]

False.

FindInstance[{m*r == (m + n)*s, r - n > s - 1, 
  r <= Max[m, n], {m, n, r} > 0}, {m, n, r, s}, Integers]

{}

It means that there are no solution satisfies the conditions.