What counts as elementary number theory, e.g., is a first course in algebraic number theory in that category for your purposes? I realize the question was asked over 2 years ago, so the original purpose for the question may well be long gone, but it'd still be worth answering if the scope of the question were clearer. Might Theorem 3 and Corollary 1 at math.uconn.edu/~kconrad/blurbs/gradnumthy/Qw2.pdf fit the condition?
– KConradMay 4 '14 at 17:10