A278572 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.

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, 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, -1, 4, 8, 14, -1

Offset

2,4

Comments

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

References

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

John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.

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

Links

Robert Israel, Table of n, a(n) for n = 2..4328 (rows 2 to 2170, flattened)

Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400. (Lists primitive trinomials only.)

Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400 [Cached copy, with permission]

R. P. Brent, Trinomial Log Files and Certificates

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.6.

Svein Mossige, Table of irreducible polynomials over GF[2] of degrees 10 through 20, Mathematics of Computation 26.120 (1972): 1007-1009.

N. Zierler and J. Brillhart, On primitive trinomials (mod 2), Information and Control 13 1968 541-554.

N. Zierler and J. Brillhart, On primitive trinomials (mod 2), II, Information and Control 14 1969 566-569.

Index entries for sequences related to trinomials over GF(2)

Example

Triangle begins:
1,
1,
1,
2,
1, 3,
1, 3,
-1,
1, 4,
3,
2,
3, 5,
-1,
5,
1, 4, 7,
-1,
3, 5, 6,
...

Maple

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
end proc:
map(T, [$2..100]); # Robert Israel, Mar 28 2017

Mathematica

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 *)

Crossrefs

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

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).

Sequence in context: A046924 A015710 A108415 * A136644 A111963 A206923

Adjacent sequences: A278569 A278570 A278571 * A278573 A278574 A278575

Keyword

sign,tabf,more

Author

N. J. A. Sloane, Nov 27 2016

Status

approved