THEOREM 2. (a) HPn immerses in R8 n-Ea(n)-3J. (b) For n even, CPn immerses in R4ln-a(n)-1]. (c) For n odd, CPn immerses in R4n-a(n). Here a(n) is the number of ones in the dyadic expansion of n, and k(n) is a non-negative function depending only on the mod (8) residue class of n with k(1) = 0, k(3) = k(5) = 1 and k(7) = 4. As a consequence, for every j>O there is some n so that RPn immerses in R2f-. From the techniques used to prove these theorems, we also obtain some results on the existence of non-singular bilinear maps.