Some new immersion results for complex projective space

We prove the following two new optimal immersion results for complex projective space. First, if n equiv 3 mod 8 but n not equiv 3 mod 64, and alpha(n)=7, then CP^n can be immersed in R^{4n-14}. Second, if n is even and alpha(n)=3, then CP^n can be immersed in R^{4n-4}. Here alpha(n) denotes the number of 1's in the binary expansion of n. The first contradicts a result of Crabb, who said that such an immersion does not exist, apparently due to an arithmetic mistake.