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Let      α  ( n ) \alpha (n)       denote the number of l’s in the binary expansion of  n . It is proved that if      n ≡  7 n \equiv 7       (8),      α  ( n ) = 6 \alpha (n) = 6      , and      n ≠  63 n \ne 63      , then      R P n {\mathbf {R}}{P^n}       can be immersed in      R 2 n −  14 {{\mathbf {R}}^{2n - 14}}      . This provides the first counterexample to the well-known conjecture that the best immersion is in      R 2 n −  2 α  ( n ) + 1 {{\mathbf {R}}^{2n - 2\alpha (n) + 1}}       (when      α  ( n ) ≡  1 \alpha (n) \equiv 1       or      2 mod 4 2 \bmod 4      ). The method of proof is obstruction theory.

The immersion conjecture for 𝑅𝑃^{8𝑙+7} is false