Liouville Random functions and normal sets

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We define a random Liouville function (λQ) which depends on a random set (Q) of primes and prove that (AQ = {n ∈ N | λQ(n) = -1 }) is normal almost everywhere. This fact enables us to generate a family of normal sets such that the equation (xy =z) is not solvable inside them. Additionally we prove that equations (xy=z², x² + y² = square, x² - y² = square) are solvable in any normal set and for any equation (xy=cn²) ((c > 1), is not a square) there exists a normal set (Ac) such that the equation is not solvable inside (Ac).