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Square-free Lucas d -pseudoprimes and Carmichael-Lucas numbers

Walter Carlip, Lawrence Somer (2007)

Czechoslovak Mathematical Journal

Let d be a fixed positive integer. A Lucas d -pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U ( P , Q ) such that the rank of N in U ( P , Q ) is exactly ( N - ε ( N ) ) / d , where ε is the signature of U ( P , Q ) . We prove here that all but a finite number of Lucas d -pseudoprimes are square free. We also prove that all but a finite number of Lucas d -pseudoprimes are Carmichael-Lucas numbers.

Squares and cubes in Sturmian sequences

Artūras Dubickas (2009)

RAIRO - Theoretical Informatics and Applications

We prove that every Sturmian word ω has infinitely many prefixes of the form UnVn3, where |Un| < 2.855|Vn| and limn→∞|Vn| = ∞. In passing, we give a very simple proof of the known fact that every Sturmian word begins in arbitrarily long squares.

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