Legendre pairs of lengths ℓ ≡ 0 (mod 5)

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Compression of complementary sequences has been proved to be a valuable tool to discover several new complementary sequences of various kinds in the past decade. Compression based search algorithms encompass a two-stage process, that typically involves the computation of several candidate compression sequences, followed by the computationally expensive decompression phase. In this paper we show how to shorten the former phase in the case of Legendre pairs of lengths ℓ≡0 (mod 5), using a plausible conjecture, supported by overwhelming numerical evidence. This is achieved by significantly decreasing the number of candidate compression sequences by using a sums-of-squares Diophantine equation. We verify our conjecture for all odd values from 3 to 17. In particular, this allows us to exhibit the first known examples of Legendre Pairs of length 85, which has been the smallest open length. We also find Legendre pairs of the open length 87 by assuming a balanced compression. As a consequence, there remains eleven lengths less than 200 for which the question of existence of a Legendre pair remains open.


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arXiv E-print repository