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How can one solve each of the equations below in positive integers?

$$2^n=5mn+7$$ $$mn2^n=5^m+7$$ $$n2^n=5^m+7$$

  • 1
    Solutions $(m,n)$: (i) $(1,5)$; (iii) $(0,2)$2012-06-04
  • 0
    Looking for all solutions2012-06-04
  • 0
    Where does this problem come from? Where is it going? Is there any reason you expect to find a general theory for solving them?2012-06-04
  • 2
    I am probing, I want to learn the different methods used to solve these equations.2012-06-04
  • 0
    This seems to be inspired by [Ramanujan's square equation](http://mathworld.wolfram.com/RamanujansSquareEquation.html) $2^n-7=x^2$.2012-06-26

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