For a certain positive integer \(m,\) the equation \(\lfloor \sqrt{n} \rfloor = m\) has 137 solutions in integers \(n.\) Find \(m.\)

n=4624; c=0; a=floor(sqrt(4624));printc," - ", n," ", a;c=c+1;n++;if(n<4761, goto2, discard=0;

Floor(137 / 2) =68 m = 68. 69^2 - 68^2 =137 4,761 - 137 =4,624, so: n=4,624

The floor of 137/2 is 68.

So m=68.

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:P