Question 1162749
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            Obviously,  this problem is  EITHER  for advanced school (?) students  OR  for not-indifferent amateurs.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;I will give only the &nbsp;<U>idea</U> &nbsp;of the proof, without going into details.



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(1)  It is "almost obvious", that the best (most closest) approximation to 4g(m) by the numbers of the form  n+n^2 = n*(n+1)  is  g(2m).


     It can be checked by analyzing inequalities . . . 



(2)  At the same time, it is easy to check manually that  g(2m)  is not equal to 4g(m).
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It is how I see the possible proof . . .