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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. 6
7056721, 79397009999, 443372888629441, 582920080863121, 2491924062668039, 14522256850701599, 39671149333495681, 242208715337316001, 729921147126771599, 842526563598720001, 1881405190466524799, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 55470688965343048319, 72631455338727028799, 122762671289519184001, 361266866679292635601, 734097107648270852639 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..20.

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.

EXAMPLE

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).

CROSSREFS

Terms that are Carmichael numbers (A002997) form A299799.

Contains A175531 as a subsequence.

Cf. A053120

Sequence in context: A253313 A253661 A088286 * A320623 A218099 A293586

Adjacent sequences: A175527 A175528 A175529 * A175531 A175532 A175533

KEYWORD

hard,nonn

AUTHOR

Max Alekseyev, Jun 08 2010

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

STATUS

approved

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Last modified January 29 00:02 EST 2023. Contains 359905 sequences. (Running on oeis4.)