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

Author

N. J. A. Sloane, Nov 27 2016

Keywords

• sign
• tabf

Comments

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

Examples

			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,
...
		

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• John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.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

Crossrefs

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

Programs

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