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All powers of 2 (Posted on 2025-03-16) Difficulty: 4 of 5
Find all positive integers (a,b,c) such that ab-c, bc-a, ca-b are all powers of 2.

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts computer findings | Comment 1 of 2
clearvars
for tot=3:2000
  for a=1:tot/3
    for b=a:(tot-a)/2
      c=tot-a-b;
      s=[a b c];
      p=prod(s);
      good=true;
      for i=1:3
        tst=p/s(i)-s(i);
        l=round(log2(tst));
        if 2^l~=tst || l<0
          good=false;
          break
        end
      end
      if good
        disp(s)
      end
    end
  end
end


finds these combinations:

     2     2     2
     2     2     3
     3     5     7
     2     6    11
 
The last two sets have three different values, in case that's an implied requirement, while the first two have members that include equal values.

  Posted by Charlie on 2025-03-16 09:43:47
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