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.

7056721, 79397009999, 443372888629441, 582920080863121, 2491924062668039, 14522256850701599, 39671149333495681, 242208715337316001, 729921147126771599, 842526563598720001, 1881405190466524799, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 55470688965343048319, 72631455338727028799, 122762671289519184001, 361266866679292635601, 734097107648270852639

Offset: 1

Author

Max Alekseyev, Jun 08 2010

Keywords

• hard
• nonn

Comments

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.

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.

No other terms below 10^21.

Named after the Russian mathematician Pafnuty Chebyshev (1821-1894) after whom the "Chebyshev polynomials" were also named. - Amiram Eldar, Jun 15 2021

Examples

			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).
		

Links

• David Broadhurst, The second Chebyshev number, NMBRTHRY Mailing List, 4 June 2010.
• Kok Seng Chua, Chebyshev polynomials and higher order Lucas Lehmer algorithm, arXiv:2010.02677 [math.NT], 2020. Mentions this sequence.
• David Pokrass Jacobs, Mohamed O. Rayes, and Vilmar Trevisan, Characterization of Chebyshev Numbers, Algebra and Discrete Mathematics, Vol. 2 (2008), pp. 65-82.
• Eric Weisstein's World of Mathematics, Lucas Sequence.

Crossrefs

Terms that are Carmichael numbers (A002997) form A299799.

Contains A175531 as a subsequence.

Cf. A053120

Extensions

a(1) is given in the Jacobs-Rayes-Trevisan paper.

a(2) from Kevin Acres, David Broadhurst, Ray Chandler, David Cleaver, Mike Oakes, and Christ van Willegen, Jun 04 2010

a(3)-a(20) from Max Alekseyev, Jun 08 2010, Feb 26 2018, Dec 16 2020