A002475 -id:A002475 - cp's OEIS frontend

A057477 Numbers k such that x^k + x^7 + 1 is irreducible over GF(2).

1, 3, 4, 6, 10, 12, 15, 18, 21, 25, 31, 34, 42, 52, 55, 57, 105, 127, 172, 210, 220, 300, 393, 420, 441, 492, 772, 807, 972, 1023, 1071, 1266, 1564, 2220, 2242, 3297, 3585, 5314, 6300, 7306, 8719, 10777, 23647, 26119, 33127, 44247, 48036, 48945, 59172, 68841

Offset: 1

Robert G. Wilson v, Sep 27 2000

nonn

The b-file contains all terms <= 300000. - Lucas A. Brown, Nov 28 2022

Lucas A. Brown, Table of n, a(n) for n = 1..58
Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

is(k) = polisirreducible(Mod(1, 2)*(x^k + x^7 + 1)); \\ Jinyuan Wang, Apr 15 2020

a(27)-a(40) from Jinyuan Wang, Apr 15 2020

a(41)-a(58) from Lucas A. Brown, Nov 28 2022

A057478 Numbers k such that x^k + x^8 + 1 is irreducible over GF(2).

Original entry on oeis.org

9, 15, 39, 105, 119, 153, 177, 209, 3143, 13169, 19833, 33567, 53129, 64439, 88871, 109865, 122945, 138543

Offset: 1

Robert G. Wilson v, Sep 27 2000

Any subsequent terms are > 300000. - Lucas A. Brown, Nov 28 2022

Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

is(k) = polisirreducible(Mod(1, 2)*(x^k + x^8 + 1)); \\ Jinyuan Wang, Apr 15 2020

a(9) from Jinyuan Wang, Apr 15 2020

a(10)-a(18) from Lucas A. Brown, Nov 28 2022

A057480 Numbers k such that x^k + x^10 + 1 is irreducible over GF(2).

Original entry on oeis.org

3, 7, 33, 111, 279, 511, 1047, 1239, 8119, 15727, 16153, 22617, 38407

Offset: 1

Robert G. Wilson v, Sep 27 2000

Any subsequent terms are > 300000. - Lucas A. Brown, Nov 28 2022

Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

is(k) = polisirreducible(Mod(1, 2)*(x^k + x^10 + 1)); \\ Jinyuan Wang, Apr 15 2020

a(7)-a(8) from Jinyuan Wang, Apr 15 2020

a(9)-a(13) from Lucas A. Brown, Nov 28 2022

A057482 Numbers n such that x^n + x^12 + 1 is irreducible over GF(2).

Original entry on oeis.org

3, 5, 7, 9, 17, 49, 97, 257, 425, 895, 1385, 4807, 11303, 25175, 103943, 104975, 161993, 282455

Offset: 1

Robert G. Wilson v, Sep 27 2000

Any subsequent terms are > 300000.

Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

is(k) = polisirreducible(Mod(1, 2)*(x^k + x^12 + 1)); \\ Jinyuan Wang, Apr 15 2020

a(10)-a(11) from Jinyuan Wang, Apr 15 2020

a(12)-a(18) from Lucas A. Brown, Nov 29 2022

A057483 Numbers n such that x^n + x^13 + 1 is irreducible over GF(2).

Original entry on oeis.org

28, 31, 33, 84, 87, 103, 174, 414, 574, 687, 780, 1111, 1449, 1860, 6964, 7708, 11700, 17428, 19398, 19876, 78391, 131305, 136564, 181684

Offset: 1

Robert G. Wilson v, Sep 27 2000

Any subsequent terms are > 300000.

Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

is(k) = polisirreducible(Mod(1, 2)*(x^k + x^13 + 1)); \\ Jinyuan Wang, Apr 15 2020

a(11)-a(14) from Jinyuan Wang, Apr 15 2020

a(15)-a(24) from Lucas A. Brown, Nov 29 2022

A058857 Numbers n such that the trinomial x^n + x + 1 is irreducible over GF(7).

Original entry on oeis.org

0, 1, 3, 4, 9, 33, 52, 177, 1042, 2799, 5950, 8595, 19438

Offset: 1

Robert G. Wilson v, Jan 05 2001

No other terms <= 4000. - Eric M. Schmidt, Feb 10 2014

Next term > 2*10^4. [Joerg Arndt, Mar 02 2016]

Cf. A002475 (GF(2)), A058334 (GF(5)).

isok(n) = polisirreducible(Mod(1, 7)*(x^n + x + 1)); \\ Michel Marcus, Feb 11 2014

def isA058857(n) : x = GF(7)['x'].0; return (x^n + x + 1).is_irreducible() # Eric M. Schmidt, Feb 10 2014

a(1), a(2) and a(10) from Eric M. Schmidt, Feb 10 2014

a(11) - a(13) from Joerg Arndt, Mar 02 2016

A074710 Numbers k such that x^k + x^2 + 1 is a primitive irreducible polynomial over GF(2).

Original entry on oeis.org

1, 3, 5, 11, 21, 29, 35, 93, 123, 333, 845

Offset: 1

Richard P. Brent, Sep 05 2002

Agrees with A057460 as far as it goes, but is a different sequence.

The next candidate is 4125.

I. F. Blake, S. Gao and R. J. Lambert, Constructive problems for irreducible polynomials over finite fields, in Information Theory and Applications, LNCS 793, Springer-Verlag, Berlin, 1994, 1-23, See Table 2.
Index entries for sequences related to trinomials over GF(2)

Cf. A002475.

A223938 Numbers n such that the trinomial x^n-x-1 is irreducible over GF(3).

Original entry on oeis.org

2, 3, 4, 5, 6, 13, 14, 17, 30, 40, 41, 51, 54, 73, 121, 137, 364, 446, 485, 638, 925, 1382, 1478, 2211, 2726, 5581, 5678, 6424, 8524, 10649, 15990, 17174, 18685, 18889, 27461, 29523, 30677, 39641, 42038, 58566, 71380, 72781, 82493

Offset: 1

Joerg Arndt, Mar 29 2013

Any subsequent terms are > 10^5. - Lucas A. Brown, Dec 11 2022

Cf. A002475 (n such that x^n-x-1 is irreducible over GF(2)).

Reap[ Do[ If[ Factor[x^n - x - 1, Modulus -> 3][[0]] =!= Times, Print[n]; Sow[n]], {n, 2, 3000}]][[2, 1]] (* Jean-François Alcover, Apr 03 2013 *)
Select[Range[1000], IrreduciblePolynomialQ[x^# - x - 1, Modulus -> 3] &] (* Robert Price, Sep 19 2018 *)

for (n=1, 10^6, if ( polisirreducible(Mod(1, 3)*(x^n-x-1)), print1(n, ", ") ) );

P. = GF(3)[]
for n in range(10^6):
       if (x^n-x-1).is_irreducible():
           print(n)

a(35)-a(43) from Lucas A. Brown, Dec 11 2022

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 *)

A057474 Numbers k such that x^k + x^5 + 1 is irreducible over GF(2).

Original entry on oeis.org

2, 3, 6, 9, 12, 14, 17, 20, 23, 44, 47, 63, 84, 129, 236, 278, 279, 297, 300, 647, 726, 737, 2574, 2660, 4233, 4500, 8207, 11900, 16046, 21983, 23999, 24596, 24849, 84929, 130926, 156308, 160046, 185142, 270641

Offset: 1

Robert G. Wilson v, Sep 27 2000

Any subsequent terms are > 300000. - Lucas A. Brown, Nov 28 2022

Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850
Lucas A. Brown, Python program.
Lucas A. Brown, Sage program.

Cf. A002475.

P. = GF(2)[]
for n in range(10^4):
    if (x^n+x^5+1).is_irreducible():
        print(n) # Joerg Arndt, Apr 28 2012

a(23)-a(34) by Joerg Arndt, Apr 28 2012
a(35)-a(38) by Manfred Scheucher, Aug 18 2015
a(39) from Lucas A. Brown, Nov 28 2022

cp's OEIS Frontend

A057477 Numbers k such that x^k + x^7 + 1 is irreducible over GF(2).

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A057478 Numbers k such that x^k + x^8 + 1 is irreducible over GF(2).

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A057480 Numbers k such that x^k + x^10 + 1 is irreducible over GF(2).

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A057482 Numbers n such that x^n + x^12 + 1 is irreducible over GF(2).

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A057483 Numbers n such that x^n + x^13 + 1 is irreducible over GF(2).

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A058857 Numbers n such that the trinomial x^n + x + 1 is irreducible over GF(7).

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PARI

Sage

Extensions

A074710 Numbers k such that x^k + x^2 + 1 is a primitive irreducible polynomial over GF(2).

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A223938 Numbers n such that the trinomial x^n-x-1 is irreducible over GF(3).

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Mathematica

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Sage

Extensions

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.