Sum of Unit Fractions Equal to 1 Has Finitely Many SolutionsMy book on abstract algebra had a small puzzle in it I was interested in. It asks the reader to prove that ∑ j = 1 n 1 x j = 1 only has finitely many solutions where x j ∈ N My first attempt to proof it was with induction. It's easy to see that for n = 1 it only has 1 solution. For the induction step I assume that in the case n = k + 1 we have infinitely many solutions, so it follows that ∑ j = 1 k 1 x j = x k − 1 x k which implies that ∑ j = 1 k 1 x j = 1 x k (assuming x k ≠ 1).This is pretty much as far as I got. I'd really love to have a hint rather than a full solution if possible.

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Sum of Unit Fractions Equal to 1 Has Finitely Many SolutionsMy book on abstract algebra had a small puzzle in it I was interested in. It asks the reader to prove that      ∑          j      =      1              n            1          x      j        =  1 only has finitely many solutions where       x    j    ∈      N  My first attempt to proof it was with induction. It&#039;s easy to see that for   n  =  1 it only has   1 solution. For the induction step I assume that in the case   n  =  k  +  1 we have infinitely many solutions, so it follows that       ∑          j      =      1              k            1          x      j        =                    x        k            −      1              x      k       which implies that       ∑          j      =      1              k            1          x      j        =      1          x      k       (assuming       x    k    ≠  1).This is pretty much as far as I got. I&#039;d really love to have a hint rather than a full solution if possible.

pararevisarii · Accepted Answer

Hint: First order the       x    i   according to size. Then try all possibilities for       x    1  , then       x    2  , etc. The constraints that       x    i    ≥      x          i      −      1       and that             n      −      i      +      1              x      i        ≥  1  −      ∑          j      =      1              i      −      1            x    j   mean that one only has to try finitely many possibilities for       x    i   at each step.

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