In the prove for Erdős–Straus Conjecture, for even \(n\), the expression,
\(\cfrac{1}{N}+\cfrac{2}{N}+\cfrac{1}{N}=\cfrac{4}{n}\)
is equivalent to
\(\cfrac{1}{N}+\cfrac{1}{N_1}+\cfrac{1}{N}=\cfrac{4}{n}\)
because \(N=n=2N_1\) is even, so the \(2\) in numerator cancel with \(N\) to give an integer \(N_1\).
If we state,
\(\cfrac{1}{A}+\cfrac{1}{B}+\cfrac{2}{C}=\cfrac{4}{n}\)
where \(n\ge 2\) and \(A\), \(B\), \(C\) are integers, it is for all \(n\) odd and even.
If we state,
\(\cfrac{1}{A}+\cfrac{1}{B}+\cfrac{1}{C}=\cfrac{4}{n}\)
then it is true, only for even integer \(n\ge 2\).
Thank you.
