let , c = b^2

now, finde natural numbers a and b which satisfies

a^c = b^a

...

This is what i did, I factoried a and b in terms of their HCF and a coprime number to g4t

a = n*p

b=n*q

Then after substituting in te above equation I got an expression, then I took 2 cases, where either the exponents of a and b are equal OR exponent of a > b's

thsat is

either b^2 = a

or b^2 > a

==> b^2 = a*n

==> b is the GM of a and n ( for any n element N and a also element N )

so one pair of soluton is a = b = 1 ..

I cant get teh others guesssing seems like abad idea even after getting b = (an)^1/2

please help, thnks!