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A175530 Pseudoprime Chebyshev numbers: odd composite integers n such that T_n(a) == a (mod n) for all integers a, where T(x) is Chebyshev polynomial of first kind.

%I A175530 #35 Feb 16 2025 08:33:12
%S A175530 7056721,79397009999,443372888629441,582920080863121,2491924062668039,
%T A175530 14522256850701599,39671149333495681,242208715337316001,
%U A175530 729921147126771599,842526563598720001,1881405190466524799,2380296518909971201,3188618003602886401,33711266676317630401,54764632857801026161,55470688965343048319,72631455338727028799,122762671289519184001,361266866679292635601,734097107648270852639
%N A175530 Pseudoprime Chebyshev numbers: odd composite integers n such that T_n(a) == a (mod n) for all integers a, where T(x) is Chebyshev polynomial of first kind.
%C A175530 Odd composite integer n is a pseudoprime Chebyshev number iff the n-th term of Lucas sequence satisfies the congruence V_n(P,1) == P (mod n) for any integer P.
%C A175530 Odd composite integer n is a pseudoprime Chebyshev number iff n == +1 or -1 (mod p-1) and n == +1 or -1 (mod p+1) for each prime p|n.
%C A175530 No other terms below 10^21.
%C A175530 Named after the Russian mathematician Pafnuty Chebyshev (1821-1894) after whom the "Chebyshev polynomials" were also named. - _Amiram Eldar_, Jun 15 2021
%H A175530 David Broadhurst, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;1d24d4ee.1006">The second Chebyshev number</a>, NMBRTHRY Mailing List, 4 June 2010.
%H A175530 Kok Seng Chua, <a href="https://arxiv.org/abs/2010.02677">Chebyshev polynomials and higher order Lucas Lehmer algorithm</a>, arXiv:2010.02677 [math.NT], 2020. Mentions this sequence.
%H A175530 David Pokrass Jacobs, Mohamed O. Rayes, and Vilmar Trevisan, <a href="http://mi.mathnet.ru/eng/adm159">Characterization of Chebyshev Numbers</a>, Algebra and Discrete Mathematics, Vol. 2 (2008), pp. 65-82.
%H A175530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>.
%e A175530 7056721 = 7 * 47 * 89 * 241, while 7056721 == 1 (mod 7-1), == 1 (mod 7+1), == -1 (mod 47-1), == 1 (mod 47+1), == 1 (mod 89-1), == 1 (mod 89+1), == 1 (mod 241-1), and == 1 (mod 241+1).
%Y A175530 Terms that are Carmichael numbers (A002997) form A299799.
%Y A175530 Contains A175531 as a subsequence.
%Y A175530 Cf. A053120
%K A175530 hard,nonn
%O A175530 1,1
%A A175530 _Max Alekseyev_, Jun 08 2010
%E A175530 a(1) is given in the Jacobs-Rayes-Trevisan paper.
%E A175530 a(2) from Kevin Acres, David Broadhurst, Ray Chandler, David Cleaver, Mike Oakes, and Christ van Willegen, Jun 04 2010
%E A175530 a(3)-a(20) from _Max Alekseyev_, Jun 08 2010, Feb 26 2018, Dec 16 2020