

A330208


Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the kth Chebyshev polynomial of the first kind.


3



5719, 6061, 11395, 15841, 17119, 18721, 31535, 67199, 73555, 84419, 117215, 133399, 133951, 174021, 181259, 194833, 226801, 273239, 362881, 469201, 516559, 522899, 534061, 588455, 665281, 700321, 721801, 778261, 903959, 1162349, 1561439, 1708901, 1755001, 1809697
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k). They noted that there are no Chebyshev pseudoprimes in both bases 2 and 3 below 2000.


LINKS



EXAMPLE

5719 is in the sequence since 5719 = 7 * 19 * 43 is composite and both T(5719 , 2)  2 and T(5719, 3)  3 are divisible by 5719.


MATHEMATICA

Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2]  2, #] && Divisible[ChebyshevT[#, 3]  3, #] &]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



