In this paper, we study rational periodic points of polynomial [Formula: see text] over the field of rational numbers, where [Formula: see text] is an integer greater than two. For period two, we describe periodic points for degrees [Formula: see text]. We also demonstrate the nonexistence of rational periodic points of exact period two for [Formula: see text] such that [Formula: see text] and [Formula: see text] has a prime factor greater than three. Moreover, assuming the [Formula: see text]-conjecture, we prove that [Formula: see text] has no rational periodic point of exact period greater than one for sufficiently large integer [Formula: see text] and [Formula: see text].
We study rational periodic points of polynomial f d,c (x) = x d + c over the field of rational numbers, where d is an integer greater than 2 and c = −1. For period 2, we classify all possible periodic points for degrees d = 4, 6, and d = 2k for positive integer k > 3 such that 2k − 1 is divisible by 3. Moreover, assuming the abc-conjecture, we prove that f d,c has no rational periodic point of exact period greater than 1 for sufficiently large integer d.
Niven’s theorem asserts that
$\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap \mathbb {Q}=\{0,\pm 1,\pm 1/2\}.$
In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set
$\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap K$
, where K is an arbitrary number field.
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