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, 3, 20, 7, -1, 5, -1, 1, 5, 14, 20, 21, -1
Offset: 2
Examples
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, ...
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- John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.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)
Crossrefs
Programs
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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
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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 *)
Comments