A278573 - OEIS

%I #36 Jun 07 2023 10:00:23

%S 1,1,2,1,3,2,3,1,3,5,1,3,4,6,-1,1,4,5,8,3,7,2,9,3,5,7,9,-1,5,9,1,4,7,

%T 8,11,14,-1,3,5,6,11,12,14,3,7,9,11,15,-1,3,5,15,17,2,7,14,19,1,21,5,

%U 9,14,18,-1,3,7,18,22,-1,-1,1,3,9,13,15,19,25,27,2,27,1,9,21,29,3,6,7,13

%N Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n-1 such x^n + x^k + 1 is irreducible (mod 2), or -1 if no such k exists.

%C Row n (if it is not -1) is invariant under the map k -> n-k. - _Robert Israel_, Mar 14 2018

%D Alanen, J. D., and Donald E. Knuth. "Tables of finite fields." Sankhyā: The Indian Journal of Statistics, Series A (1964): 305-328.

%D John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.

%D Marsh, Richard W. Table of irreducible polynomials over GF (2) through degree 19. Office of Technical Services, US Department of Commerce, 1957.

%H Robert Israel, <a href="/A278573/b278573.txt">Table of n, a(n) for n = 2..4558</a> (rows 2 to 1300, flattened)

%H Joerg Arndt, <a href="http://www.jjj.de/mathdata/all-trinomial-primpoly.txt">Complete list of primitive trinomials over GF(2) up to degree 400</a>. (Lists primitive trinomials only.)

%H Joerg Arndt, <a href="/A001153/a001153.txt">Complete list of primitive trinomials over GF(2) up to degree 400</a> [Cached copy, with permission]

%H R. P. Brent, <a href="http://maths.anu.edu.au/~brent/trinomlg.html">Trinomial Log Files and Certificates</a>

%H A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://www.cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, CRC Press, 1996; see Table 4.6.

%H Svein Mossige, <a href="https://doi.org/10.1090/S0025-5718-1972-0313227-5">Table of irreducible polynomials over GF[2] of degrees 10 through 20</a>, Mathematics of Computation 26.120 (1972): 1007-1009.

%H N. Zierler and J. Brillhart, <a href="http://dx.doi.org/10.1016/S0019-9958(68)90973-X">On primitive trinomials (mod 2)</a>, Information and Control 13 1968 541-554.

%H N. Zierler and J. Brillhart, <a href="http://dx.doi.org/10.1016/S0019-9958(69)90356-8">On primitive trinomials (mod 2), II</a>, Information and Control 14 1969 566-569.

%H <a href="/index/Tri#trinomial">Index entries for sequences related to trinomials over GF(2)</a>

%e Triangle begins:

%e 1,

%e 1, 2,

%e 1, 3,

%e 2, 3,

%e 1, 3, 5,

%e 1, 3, 4, 6,

%e -1,

%e 1, 4, 5, 8,

%e 3, 7,

%e 2, 9,

%e 3, 5, 7, 9,

%e -1,

%e 5, 9,

%e 1, 4, 7, 8, 11, 14,

%e -1,

%e 3, 5, 6, 11, 12, 14,

%e 3, 7, 9, 11, 15,

%e -1,

%e 3, 5, 15, 17,

%e 2, 7, 14, 19,

%e 1, 21,

%e ...

%p for n from 2 to 30 do

%p   S:= select(k -> Irreduc(x^n+x^k+1) mod 2, [$1..n-1]);

%p   if S = [] then print(-1) else print(op(S)) fi

%p od: # _Robert Israel_, Mar 14 2018

%Y Cf. A001153, A057646, A057774, A073571, A073646, A073726, A074743, A278572.

%K sign,tabf

%O 2,3

%A _N. J. A. Sloane_, Nov 27 2016