Math Olympiad | Wonderful Exponential Problem | 90% Failed to solve!

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[@sy8146]

Thank you for explaining. There is another solution. It is "m=18, n=1" .

[@vijaymaths5483]

@@sy8146 Most welcome

[@marcgriselhubert3915]

The fact that the second solution m = 18 with n = 1 is not found with this method is sufficient to say that this method is false.
x = m^(n/2), y = n^(m/2), x + y and x - y are not necessary integers, that's the problem.
This problem is difficult to solve correctly. With inequations it is possible to prove that n and m have limited possibilities, then it is sufficient to test one by one these possibilities.

[@RealQinnMalloryu4]

(n ➖ 1m+1).