A note on Chebyshev polynomials, cyclotomic polynomials and twin primes

We revisit Stephen P. Humphriesʼ results indicating some connections between Chebyshev polynomials and twin primes, by using Chebyshev polynomials of the third and fourth kinds and cyclotomic polynomials. We then give counterexamples to a conjecture of Humphriesʼ. We also remark another characterization of twin primes in terms of Chebyshev polynomials of the second kind.