A057646 -id:A057646 - cp's OEIS frontend

A073571 Irreducible trinomials: numbers n such that x^n + x^k + 1 is an irreducible polynomial (mod 2) for some k with 0 < k < n.

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 20, 21, 22, 23, 25, 28, 29, 30, 31, 33, 34, 35, 36, 39, 41, 42, 44, 46, 47, 49, 52, 54, 55, 57, 58, 60, 62, 63, 65, 66, 68, 71, 73, 74, 76, 79, 81, 84, 86, 87, 89, 90, 92, 93, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108, 110, 111, 113

Offset: 1

Paul Zimmermann, Sep 05 2002

nonn

This sequence is infinite: Golomb, "Shift Register Sequences," on p. 96 (1st ed., 1966) states that "It is easy to exhibit an infinite class of irreducible trinomials. viz. x^(2*3^a) + x^(3^a) + 1 for all a = 0, 1, 2, ..., but whose roots have only 3^(a+1) as their period." - A. M. Odlyzko, Dec 05 1997.

S. W. Golomb, "Shift register sequence", revised edition, reprinted by Aegean Park Press, 1982. See Tables V-1, V-2.

Joerg Arndt, Table of n, a(n) for n = 1..1500
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.6.
Index entries for sequences related to trinomials over GF(2)

For the numbers of such trinomials for a given n, see A057646.

See A073726 for primitive trinomials and A001153 for primitive Mersenne trinomials (and references). Complement of A057486. For values of k see A057774.

a := proc(n) local k; for k from 1 to n-1 do if Irreduc(x^n+x^k+1) mod 2 then RETURN(n) fi od; NULL end: [seq(a(n), n=1..130)];

irreducibleQ[n_] := (irr = False; k = 1; While[k < n, If[ Factor[ x^n + x^k + 1, Modulus -> 2] == x^n + x^k + 1, irr = True; Break[]]; k++]; irr); Select[ Range[120], irreducibleQ] (* Jean-François Alcover, Jan 07 2013 *)

is(n)=for(s=1,n-1,if(polisirreducible((x^n+x^s+1)*Mod(1,2)), return(1)));0 \\ Charles R Greathouse IV, May 30 2013

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.

Original entry on oeis.org

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

N. J. A. Sloane, Nov 27 2016

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

			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,
...

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.

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)

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).

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

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 *)

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.

Original entry on oeis.org

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

N. J. A. Sloane, Nov 27 2016

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

			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,
...

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.

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)

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

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

A344146 a(n) is the number of pentanomials x^n + x^a + x^b + x^c + 1 that are irreducible over GF(2) for n > a > b > c > 0.

Original entry on oeis.org

1, 4, 6, 10, 17, 22, 38, 46, 54, 66, 73, 98, 94, 152, 124, 158, 199, 184, 226, 296, 202, 406, 328, 334, 418, 380, 486, 584, 351, 666, 578, 658, 896, 604, 728, 964, 577, 1128, 925, 846, 1286, 898, 1102, 1520, 760, 1628, 1421, 1312, 1837, 1298, 1580, 2220, 1142, 2346, 1764, 1524, 2782

Offset: 4

Jianing Song, May 10 2021

It is conjectured that a(n) > 0 for all n >= 4.

			a(4) = 1 because there is only one irreducible pentanomial of degree 4 over GF(2), namely x^4 + x^3 + x^2 + x + 1.
a(6) = 4 because there are 4 irreducible pentanomials of degree 6 over GF(2): x^6 + x^4 + x^2 + x + 1, x^6 + x^4 + x^3 + x + 1, x^6 + x^5 + x^2 + x + 1, x^6 + x^5 + x^3 + x^2 + 1, x^6 + x^5 + x^4 + x + 1 and x^6 + x^5 + x^4 + x^2 + 1.
a(7) = 10 since the 10 irreducible pentanomials of degree 6 over GF(2) are of the form x^7 + x^a + x^b + x^c + 1 for (a,b,c) = (3,2,1), (4,3,2), (5,2,1), (5,3,1), (5,4,3), (6,3,1), (6,4,1), (6,4,2), (6,5,2), (6,5,4).

Jianing Song, Table of n, a(n) for n = 4..173

Cf. A014580 (irreducible polynomials over GF(2) encoded as binary numbers), A057646.

a(n) = sum(a=3, n-1, sum(b=2, a-1, sum(c=1, b-1, polisirreducible(Mod(x^n+x^a+x^b+x^c+1, 2)))))

cp's OEIS Frontend

A073571 Irreducible trinomials: numbers n such that x^n + x^k + 1 is an irreducible polynomial (mod 2) for some k with 0 < k < n.

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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.

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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.

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A344146 a(n) is the number of pentanomials x^n + x^a + x^b + x^c + 1 that are irreducible over GF(2) for n > a > b > c > 0.

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