A note on Chebyshev polynomials, cyclotomic polynomials and twin primes

The class of Lucas-Lehmer polynomials

“…Finally, it is well known that Chebyshev polynomials can be applied in several fields of Mathematics, for example in Numerical Analysis and Combinatorics (see, for instance: [3,5,7,13,19,23,26,4] and references therein). We think that also the Lucas-Lehmer polynomials could be adapted to solve similar problems.…”

“…In this paper we show some properties of these polynomials, in particular discussing the link among the Lucas-Lehmer polynomials and the Chebyshev polynomials of the first and second kind [2,12,22]. The Chebyshev polynomials are well-known and, although they have been known and studied for a long time, continue to play an important role in recent advances in many areas of mathematics such as Algebra, Numerical Analysis, Differential Equations and Number Theory (see, for instance: [1,3,5,7,10,13,19,23,26,4]) and new other properties of theirs continue to be discovered ([3, 7, 8, 11]).…”

Generalized Markoff equations and Chebyshev polynomials

“…The following characterization shows a connection between Chebyshev polynomials of the second kind and twin primes (see [Yam13] for details). The following statements are equivalent:…”

“…From before, we know that the tree E(1, 2, 3) (which appears in [Coh79] and [Zag82]) consists of all triples…”

“…Each x * descends (uniquely) to (1, 2, 3). Thus, for any integer n > 2, there are 1 2 ϕ(n) triples x * in E(1, 2, 3) so that x * 3 = n, where ϕ is the Euler totient function (see [Zag82] for details).…”