Homotopy exponents of Moore spaces

We prove that 2 r + 1 ⋅ π ∗ ( P m ( 2 r ) ) = 0 provided m ≥ 4 and r ≥ 6 . This is the best possible result. As well, for 2 ≤ r ≤ 5 we obtain upper bounds on the homotopy exponent of P m ( 2 r ) .