Legendre pairs of lengths $\ell \equiv 0$ (mod $3$)

We prove a proposition that connects constant-PAF sequences and the corresponding Legendre pairs with integer PSD values. We show how to determine explicitly the complete spectrum of the $(\ell/3)$-rd value of the discrete Fourier transform for Legendre pairs of lengths $\ell \equiv 0 \, (\mod 3)$. This is accomplished by two new algorithms based on number-theoretic arguments. As an application, we prove that Legendre pairs of the open lengths 117, 129, 133, and 147 exist by finding Legendre pairs of these lengths with a multiplier group of order at least 3. As a consequence, 85, 87, 115, 145, 159, 161, 169, 175, 177, 185, 187, 195 are the twelve integers in the range < 200 for which the question of existence of Legendre pairs remains unsolved.