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Legendre pairs of lengths $\ell\equiv0$ (mod 5)

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arxiv 2111.02105 v3 pith:JEL6JP7Z submitted 2021-11-03 math.CO math.NT

Ilias Kotsireas , Christoph Koutschan , Dursun Bulutoglu , David Arquette , Jonathan Turner , Kenneth Ryan This is my paper · ORCID
classification math.COmath.NT
keywords legendrelengthlengthspairsdecreaseassumingbalanceconjecture
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abstract

By assuming a type of balance for length $\ell=87$ and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length $\ell=85$, we find LPs of these lengths. We then study the power spectral density (PSD) values of m-compressions of LPs of length 5m. We also formulate a conjecture for Legendre pairs of lengths $\ell \equiv 0$ (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range $\leq 200$ for which the existence question of LPs remains unsolved from 12 to 10.

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  1. Multiplier obstructions for Legendre pairs of length 333

    math.CO 2026-07 accept novelty 7.0 of 10

    A Legendre pair of length 333 cannot be invariant under any common multiplier subgroup of order at least 9; only subgroups of order at most 6 remain possible.

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