A278572 - OEIS

%I #52 Jun 07 2023 10:29:44

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

%T 1,5,9,-1,3,7,-1,-1,1,3,9,13,2,1,9,3,6,7,13,-1,10,13,7,2,9,11,15,-1,

%U -1,4,8,14,-1

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

%C This is the format used by John Brillhart (1968) and Zierler and Brillhart (1968).

%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%D John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.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="/A278572/b278572.txt">Table of n, a(n) for n = 2..4328</a> (rows 2 to 2170, 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,

%e 1,

%e 2,

%e 1, 3,

%e 1, 3,

%e -1,

%e 1, 4,

%e 3,

%e 2,

%e 3, 5,

%e -1,

%e 5,

%e 1, 4, 7,

%e -1,

%e 3, 5, 6,

%e ...

%p T:= proc(n) local L; L:= select(k -> Irreduc(x^n+x^k+1) mod 2, [$1..n/2]); if L = [] then -1 else op(L) fi

%p end proc:

%p map(T, [$2..100]); # _Robert Israel_, Mar 28 2017

%t DeleteCases[#, 0] & /@ Table[Boole[IrreduciblePolynomialQ[x^n + x^# + 1, Modulus -> 2]] # & /@ Range[Floor[n/2]], {n, 2, 40}] /. {} -> {-1} // Flatten (* _Michael De Vlieger_, Mar 28 2017 *)

%Y Cf. A001153, A057646, A057774, A073571, A073646, A073726, A074743, A278573.

%Y Rows n that contain particular numbers: 1 (A002475), 2 (A057460), 3 (A057461), 4 (A057463), 5 (A057474), 6 (A057476), 7 (A057477), 8 (A057478), 9 (A057479), 10 (A057480), 11 (A057481), 12 (A057482), 13 (A057483).

%K sign,tabf,more

%O 2,4

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