Some new immersion results for complex projective space

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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 CPⁿ can be immersed in R⁴ⁿ⁻¹⁴. Second, if n is even and alpha(n)=3, then CPⁿ can be immersed in R⁴ⁿ⁻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.