We prove a proposition that connects constant-periodic autocorrelation function sequences and the corresponding Legendre pairs with integer power spectral density values. We show how to determine explicitly the complete spectrum of the $\left(\ell ∕3\right)$-rd value of the discrete Fourier transform for Legendre pairs of lengths $\ell \equiv 0\left(\mathrm{mod}3\right)$. 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 12 integers in the range $<200$ for which the question of existence of Legendre pairs remains unsolved.

中文翻译：

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勒让德长度对 ℓ ≡ 0 ( mod 3 )

我们证明了一个命题，该命题将恒定周期自相关函数序列和相应的勒让德对与整数功率谱密度值连接起来。我们展示了如何明确地确定$\left(\ell ∕3\right)$-Legendre 长度对的离散傅立叶变换的 -rd 值 $\ell \equiv 0\left(\mathrm{模组}3\right)$. 这是通过两种基于数论论证的新算法实现的。作为一个应用，我们通过找到具有至少 3 阶乘数群的这些长度的勒让德对来证明这些开放长度的勒让德对存在。因此，85、87、115、145 , 159, 161, 169, 175, 177, 185, 187, 195 是范围内的 12 个整数$<200$ 对此，勒让德对的存在问题仍未解决。