A278573 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.

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, 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, 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

Offset

2,3

Comments

Row n (if it is not -1) is invariant under the map k -> n-k. - Robert Israel, Mar 14 2018

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John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.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..4558 (rows 2 to 1300, 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, 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, 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,
...

Maple

for n from 2 to 30 do
  S:= select(k -> Irreduc(x^n+x^k+1) mod 2, [$1..n-1]);
  if S = [] then print(-1) else print(op(S)) fi
od: # Robert Israel, Mar 14 2018

Crossrefs

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

Sequence in context: A318928 A159918 A349552 * A108663 A379631 A307314

Adjacent sequences: A278570 A278571 A278572 * A278574 A278575 A278576

Keyword

sign,tabf

Author

N. J. A. Sloane, Nov 27 2016

Status

approved