do there exist integers nm N Do there exists integers n m N

do there exist integers n,m N Do there exists integers n, m N with n, m > 1 such that n^5 = m^2 , where n is square-free (i.e. no perfect square divides n)?

Solution

No . if n is square free then its prime factorisation must have at least one number which is not paird ex.

in 12 = 2x 2x 3, 3 is not paired hence 12 is not perfect square. call this number t. in n^5 we get the factor t^5 so even after pairing we get one t left . but in right hand side . we shal get exact pairs of factors because it is a square. hence not possible.(we implictly used the prime factorisation theorem that prime factor of a number is unique if 1 is considered once).