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A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Let f ( x ) = x ⁿ + k n - 1 x n - 1 + k n - 2 x n - 2 + ⋯ + k ₁ x + k ₀ ∈ ℤ [ x ] , where 3 ≤ k n - 1 ≤ k n - 2 ≤ ⋯ ≤ k ₁ ≤ k ₀ ≤ 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.

A class of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Let q > 2 be a prime power and f = - x + t x q + x 2 q - 1 , where t ∈ * q . We prove that f is a permutation polynomial of q ² if and only if one of the following occurs: (i) q is even and T r q / 2 ( 1 / t ) = 0 ; (ii) q ≡ 1 (mod 8) and t² = -2.

A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm

Jun-ichi Tamura (1992)

Acta Arithmetica

In this paper, we give transcendental numbers φ and ψ such that (i) both φ and ψ have explicit g-adic expansions, and simultaneously, (ii) the vector t ( φ , ψ ) has an explicit expression in the Jacobi-Perron algorithm (cf. Theorem 1). Our results can be regarded as a higher-dimensional version of some of the results in [1]-[5] (see also [6]-[8], [10], [11]). The numbers φ and ψ have some connection with algebraic numbers with minimal polynomials x³ - kx² - lx - 1 satisfying (1.1) k ≥ l ≥0, k + l ≥ 2 (k,l...

A class of weakly perfect graphs

H. R. Maimani, M. R. Pournaki, S. Yassemi (2010)

Czechoslovak Mathematical Journal

A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given.

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