How do I make Mathematica show me the solutions that it has found while it is searching for them

Question

How do I make Mathematica show me the solutions that it has found while it is searching for them, specifically in this:

Solve[
  {{(n^2) (s - 2) - (n) (s - 4)}/2 == 2^p - 1, 2 < n < 2100,0 < p < 10, 9 < s < 21}, 
  {n, p, s}, Integers]

The ranges can/will be adjusted, I just want to set it so that I can see the progress that Mathematica is making.

Answer 1

With a single Solve command, I think this is impossible. But you can do something like the following. There are some strange curly brackets in your equation, so I hope I repaired it correctly. (I did not find a solution, and FindInstance did not give a solution either.)

Monitor[Table[{p,s, sol=Solve[(n^2 (s-2)-n (s-4))/2==2^p-1 && 2<n<2100, n,Integers] }, {p,1,9},{s, 10, 20}], {p,s,sol}]

Monitoring does not show much here, because the result turns up almost immediately.

Instead of Monitor, you can use a Print command inside the Table:

Table[sol=Solve[(n^2 (s-2)-n (s-4))/2==2^p-1 && 2<n<2100, n,Integers];
  Print[{p,s,sol}]; {p,s,sol}, {p,1,9},{s,10, 20}]

If you do not want to see the empty results, you can do something like

result = {};
Do[sol = Solve[(n^2 (s - 2) - n (s - 4))/2 == 2^p - 1 && 2 < n < 2100,
   n, Integers];
   If[sol != {}, Print[{p, s, sol}]; 
   AppendTo[result, {p, s, sol}]], {p, 1, 9}, {s, 10, 20}];
result

You might place the Do-loop in a Monitor, replace the Print statement with qq={p,s,sol} and use qq in the second argument of Monitor.