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
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
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