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Primefree shifted Lucas sequences

Lenny Jones — 2015

Acta Arithmetica

We say a sequence = ( s ) n 0 is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, a = ( u ) n 0 , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences a ± k are simultaneously primefree. This result extends...

Generalizing a theorem of Schur

Lenny Jones — 2014

Czechoslovak Mathematical Journal

In a letter written to Landau in 1935, Schur stated that for any integer t > 2 , there are primes p 1 < p 2 < < p t such that p 1 + p 2 > p t . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers t k 1 and real ϵ > 0 , there exist primes p 1 < p 2 < < p t such that p 1 + p 2 + + p k > ( k - ϵ ) p t .

A class of irreducible polynomials

Joshua HarringtonLenny Jones — 2013

Colloquium Mathematicae

Let f ( x ) = x + k n - 1 x n - 1 + k n - 2 x n - 2 + + k x + k [ x ] , where 3 k n - 1 k n - 2 k k 2 k n - 1 - 3 . We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of 2 k n - 1 - 3 on the coefficients of f(x) is the best possible in this situation.

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