A073571 -id:A073571 - cp's OEIS frontend

A057774 Values of k corresponding to A073571.

1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 5, 1, 3, 3, 3, 2, 1, 5, 3, 1, 2, 1, 3, 10, 7, 2, 9, 4, 3, 7, 5, 1, 5, 9, 3, 9, 7, 4, 19, 1, 29, 1, 18, 3, 9, 6, 25, 35, 21, 9, 4, 5, 21, 13, 38, 27, 21, 2, 21, 11, 6, 11, 15, 29, 9, 4, 15, 17, 33, 10, 9, 33, 8, 18, 2, 19, 21, 1, 5, 3

Offset: 1

N. J. A. Sloane, Sep 06 2002

nonn

If there is more than one choice for k then the smallest value is taken.

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.6.

A073726 Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 15, 17, 18, 20, 21, 22, 23, 25, 28, 29, 31, 33, 35, 36, 39, 41, 47, 49, 52, 55, 57, 58, 60, 63, 65, 68, 71, 73, 79, 81, 84, 87, 89, 93, 94, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 118, 119, 121, 123, 124, 127, 129, 130, 132, 134, 135, 137, 140, 142, 145, 148, 150, 151, 153, 159, 161, 167, 169, 170, 172, 174, 175, 177, 178, 183, 185, 191, 193, 194, 198, 199, 201

Offset: 1

Richard P. Brent and Paul Zimmermann, Sep 05 2002

Start is similar to A194125; first terms here but missing there are 140, 212, 236.

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

Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400.
Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400 [Cached copy, with permission]
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.8.
Index entries for sequences related to trinomials over GF(2)

See A073571 for irreducible trinomials and A001153 for primitive Mersenne trinomials (and references). See A074744 for values of k.

Cf. A194125 (n such that x^n+(1+x)^w over GF(2) is primitive for some w).

A073726 := function(n) for k := 1 to n-1 do if IsPrimitive(x^n+x^k+1) then return true; end if; end for; return false; end function; l := []; for n := 1 to 100 do if A073726(n) then l := Append(l,n); end if; end for; l;

A073726 := proc(n) local k,m: option remember: if(n=1)then return 2: else m:=procname(n-1)+1: while(true)do for k from 1 to m-1 do if Primitive(x^m+x^k+1) mod 2 then return m: fi: od: m:=m+1: od: fi: end:
seq(A073726(n),n=1..20); # Nathaniel Johnston, Apr 26 2011

okQ[n_] := AnyTrue[Range[n-1], PrimitivePolynomialQ[x^n + x^# + 1, 2]&];
Select[Range[201], okQ] (* Jean-François Alcover, Aug 19 2019 *)

a(49)-a(58) from Nathaniel Johnston, Apr 26 2011

A001153 Degrees of primitive irreducible trinomials: n such that 2^n - 1 is a Mersenne prime and x^n + x^k + 1 is a primitive irreducible polynomial over GF(2) for some k with 0 < k < n.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 89, 127, 521, 607, 1279, 2281, 3217, 4423, 9689, 19937, 23209, 44497, 110503, 132049, 756839, 859433, 3021377, 6972593, 24036583, 25964951, 30402457, 32582657, 42643801, 43112609

Offset: 1

N. J. A. Sloane

Also the list of "irreducible Mersenne trinomials" since here irreducible implies primitive.
Further terms of the form +-3 (mod 8) are unlikely, as the only possibility of an irreducible trinomial for n == +-3 (mod 8) is (by Swan's theorem) x^n+x^2+1 (and its reciprocal); see the Ciet et al. and the Swan reference. - Joerg Arndt, Jan 06 2014
The first Mersenne prime exponent not ruled out by Swan's theorem and yet not a member of this sequence is 57885161. - Gord Palameta, Jul 20 2018
74207281 is also in the sequence. - Gord Palameta, Jul 20 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Matters Computational (The Fxtbook), see p.850 (but note errata for statement of Swan's theorem).
Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400.
Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400 [Cached copy, with permission]
R. P. Brent, Searching for primitive trinomials (mod 2) (first, "old" page)
R. P. Brent, Searching for primitive trinomials (mod 2) (second, current page)
R. P. Brent, Trinomial Log Files and Certificates
Richard P. Brent and Paul Zimmerman, Twelve New Primitive Binary Trinomials, HAL Id : hal-01378493.
R. P. Brent, S. Larvala and P. Zimmermann, A fast algorithm for testing reducibility of trinomials ..., Math. Comp. 72 (2003), 1443-1452.
Mathieu Ciet, Jean-Jacques Quisquater, Francesco Sica, A Short Note on Irreducible Trinomials in Binary Fields, in: 23rd Symposium on Information Theory in the BENELUX, Louvain-la-Neuve, Belgium, Macq, B., Quisquater, J.-J. (eds.), pp.233-234, (May-2002).
Yoshiharu Kurita and Makoto Matsumoto, Primitive t-nomials (t=3,5) over GF(2) whose degree is a Mersenne exponent <= 44497, Math. Comp. 56 (1991), no. 194, 817-821.
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 162.
Richard G. Swan, Factorization of polynomials over finite fields, Pacific Journal of Mathematics, vol.12, no.3, pp.1099-1106, (1962).
N. Zierler, Primitive trinomials whose degree is a Mersenne exponent, Information and Control 15 1969 67-69.
N. Zierler, On x^n+x+1 over GF(2), Information and Control 16 1970 502-505.
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. A002475, A000043, A073571, A073639, A057486, A073726.

For smallest values of k, see A074743.

Corrected and extended by Paul Zimmermann, Sep 05 2002

Six more terms from Brent's page added by Max Alekseyev, Oct 22 2011

A073639 Numbers k such that x^k + x + 1 is a primitive polynomial modulo 2.

Original entry on oeis.org

2, 3, 4, 6, 7, 15, 22, 60, 63, 127, 153, 471, 532, 865, 900, 1366

Offset: 1

Richard P. Brent and Paul Zimmermann, Sep 05 2002

Subsequence of A002475, which gives k for which the polynomial x^k + x + 1 is irreducible modulo 2. Term m of A002475 belongs to this sequence iff A046932(m) = 2^m - 1.
Note that a(16) = 1366 = A002475(23). For k = A002475(24) and A002475(25), polynomial x^k + x + 1 is not primitive modulo 2, so a(17) >= A002475(26) = 4495.
The following large terms of A002475 do not belong here: 53484, 62481, 83406, 103468. - Max Alekseyev, Aug 18 2015

Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850
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.
R. P. Brent, Searching for primitive trinomials (mod 2)
R. P. Brent, S. Larvala and P. Zimmermann, A fast algorithm for testing reducibility of trinomials ..., Math. Comp. 72 (2003), 1443-1452.
N. Zierler, Primitive trinomials whose degree is a Mersenne exponent, Information and Control 15 1969 67-69.
N. Zierler, On x^n+x+1 over GF(2), Information and Control 16 1970 502-505.
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. A002475, A073571, A057486.

Select[Range[2, 1000], PrimitivePolynomialQ[x^# + x + 1, 2] &] (* Robert Price, Sep 19 2018 *)

A057486 Numbers k such that x^k + x^m + 1 is factorable over GF(2) for all m between 1 and k.

Original entry on oeis.org

8, 13, 16, 19, 24, 26, 27, 32, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 59, 61, 64, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 91, 96, 99, 101, 104, 107, 109, 112, 114, 115, 116, 117, 120, 122, 125, 128, 131, 133, 136, 138, 139, 141, 143, 144, 149, 152, 157

Offset: 1

Robert G. Wilson v, Sep 28 2000

nonn

Brent, Hart, Kruppa, and Zimmermann found that 57885161 is a term of this sequence. - Charles R Greathouse IV, May 30 2013

			a(1) = 8 because
x^8 + x^1 + 1 = (1 + x + x^2)*(1 + x^2 + x^3 + x^5 + x^6),
x^8 + x^2 + 1 = (1 + x + x^4)^2,
x^8 + x^3 + 1 = (1 + x + x^3)*(1 + x + x^2 + x^3 + x^5),
x^8 + x^4 + 1 = (1 + x + x^2)^4,
x^8 + x^5 + 1 = (1 + x^2 + x^3)*(1 + x^2 + x^3 + x^4 + x^5),
x^8 + x^6 + 1 = (1 + x^3 + x^4)^2, and
x^8 + x^7 + 1 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^6).

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 200 terms from T. D. Noe)
Richard Brent, The Software gf2x.
Paul Zimmermann, There is no primitive trinomial of degree 57885161 over GF(2), posting to NMBRTHRY mailing list [alternate link]
Index entries for sequences related to trinomials over GF(2)

Complement of A073571. Cf. A001153, A002475, A073639.

Do[ k = 1; While[ ToString[ Factor[ x^n + x^k + 1, Modulus -> 2 ]] != ToString[ x^n + x^k + 1 ] && k < n, k++ ]; If[ k == n, Print[ n ]], {n, 2, 234} ]

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

A057646 a(n) is the number of trinomials x^n + x^k + 1 that are irreducible over GF(2) for some k with n > k > 0.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 0, 4, 2, 2, 4, 0, 2, 6, 0, 6, 5, 0, 4, 4, 2, 4, 0, 4, 0, 0, 8, 2, 4, 8, 0, 4, 2, 2, 6, 0, 0, 6, 0, 4, 2, 0, 2, 0, 2, 8, 0, 8, 0, 0, 8, 0, 5, 4, 0, 8, 2, 0, 12, 0, 2, 10, 0, 4, 2, 0, 4, 0, 0, 10, 0, 6, 2, 0, 2, 0, 0, 4, 0, 6, 0, 0, 14, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 4, 0, 8, 4, 0, 10

Offset: 2

Robert G. Wilson v, Oct 11 2000

Brent, Hart, Kruppa, and Zimmermann found that a(57885161) = 0. - Charles R Greathouse IV, May 30 2013

			a(7) = 4 because 1 + x + x^7 = 1 + x + x^7, 1 + x^2 + x^7 = (1 + x + x^2)*(1 + x + x^2 + x^4 + x^5), 1 + x^3 + x^7 = 1 + x^3 + x^7, 1 + x^4 + x^7 = 1 + x^4 + x^7, 1 + x^5 + x^7 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^5) and 1 + x^6 + x^7 = 1 + x^6 + x^7. Thus there are 4 trinomial expressions which cannot be factored over GF(2) and 2 trinomial expressions which do factor.

T. D. Noe, Table of n, a(n) for n = 2..500
Paul Zimmermann, There is no primitive trinomial of degree 57885161 over GF(2), posting to NMBRTHRY mailing list [alternate link]

For n such that a(n) > 0 see A073571.

Cf. A014580 (irreducible polynomials over GF(2) encoded as binary numbers), A344146.

a(n)=sum(s=1,n-1,polisirreducible((x^n+x^s+1)*Mod(1,2))) \\ Charles R Greathouse IV, May 30 2013

A344142 Lexicographically first irreducible polynomial over GF(2) of degree n with the lowest possible number of terms, evaluated at X = 2.

Original entry on oeis.org

2, 7, 11, 19, 37, 67, 131, 283, 515, 1033, 2053, 4105, 8219, 16417, 32771, 65579, 131081, 262153, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108891, 134217767, 268435459, 536870917, 1073741827, 2147483657, 4294967437, 8589935617

Offset: 1

Jianing Song, May 10 2021

nonn

Different from A344141, here you first check x^n + x + 1, x^n + x^2 + 1, ..., x^n + x^(n-1) + 1 until you get an irreducible polynomial over GF(2); if there are none, you then check x^n + x^3 + x^2 + x + 1, x^n + x^4 + x^2 + x + 1, x^n + x^4 + x^3 + x + 1, x^n + x^4 + x^3 + x^2 + 1, ..., x^n + x^(n-1) + x^(n-2) + x^(n-3) + 1 until you get an irreducible polynomial over GF(2). Once you find it, evaluate it at x = 2.
Note that it is conjectured that an irreducible polynomial of degree n with 5 terms exists for every n. It follows from the conjecture that A000120(a(n)) = 3 for n in A073571 and 5 for n in A057486.
In A057496 it is stated that if x^n + x^3 + x^2 + x + 1 is irreducible, then so is x^n + x^3 + 1. It follows that no term other than 19 can be of the form 2^n + 15.

			a(33) = 8589935617, since x^33 + x + 1, x^33 + x^2 + 1, x^33 + x^3 + 1, ..., x^33 + x^9 + 1 are all reducible over GF(2) and x^33 + x^10 + 1 is irreducible, so a(33) = 2^33 + 2^10 + 1 = 8589935617.
a(8) = 283, since x^8 + x + 1, x^8 + x^2 + 1, ..., x^8 + x^7 + 1 are all reducible over GF(2); both x^8 + x^3 + x^2 + x + 1, x^8 + x^4 + x^2 + x + 1 are reducible, and x^8 + x^4 + x^3 + x + 1 is irreducible, so a(8) = 2^8 + 2^4 + 2^3 + 2 + 1 = 283.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A014580, A344141, A344143, A000120, A073571, A057486, A057496.

A344142(n) = if(n==1, 2, for(k=1, n-1, if(polisirreducible(Mod(x^n+x^k+1, 2)), return(2^n+2^k+1))); for(a=3, n-1, for(b=2, a-1, for(c=1, b-1, if(polisirreducible(Mod(x^n+x^a+x^b+x^c+1, 2)), return(2^n+2^a+2^b+2^c+1)))))) \\ Assuming that an irreducible polynomial of degree n with at most 5 terms exists for every n.

A272486 Numbers n > 1 such that the polynomial x^(n-k)*(x+1)^k+1 over GF(2) is reducible for all k from 1 to n-1.

Original entry on oeis.org

8, 13, 16, 19, 24, 26, 27, 32, 37, 38, 40, 42, 43, 45, 48, 50, 51, 53, 54, 56, 59, 61, 64, 66, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 90, 91, 96, 99, 101, 104, 107, 109, 110, 112, 114, 115, 116, 117, 120, 122, 125, 126, 128, 131, 133, 136, 138, 139

Offset: 1

Luis H. Gallardo, May 01 2016

nonn

If n = p is an odd prime, the sequence appears coincide with A057749. Is this true, and if so, why?

More generally, the sequence appears the same as the set complement of A073571 (minus {1}). Again, if true, why? [This is not correct. E.g., the terms 42, 54, 66, 90, and 110 are in both sequences. Joerg Arndt, Apr 29 2019]

Cf. A057749, A073571.

ok[n_] := {} == Quiet@ Select[Range[n-1], IrreduciblePolynomialQ[ x^(n-#) * (x+1)^# + 1, Modulus -> 2] &, 1]; Select[Range[2, 140], ok] (* Giovanni Resta, May 02 2016 *)

isok(n) = if (n<=1, 0, for (k=1, n-1, if (polisirreducible(Mod(1,2)*(x^(n-k)*(x+1)^k+1)), return(0));); 1;); \\ Michel Marcus, May 02 2016

a(41)-a(60) from Giovanni Resta, May 02 2016

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

cp's OEIS Frontend

A057774 Values of k corresponding to A073571.

Views

Author

Keywords

Comments

Links

A073726 Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Extensions

A001153 Degrees of primitive irreducible trinomials: n such that 2^n - 1 is a Mersenne prime and x^n + x^k + 1 is a primitive irreducible polynomial over GF(2) for some k with 0 < k < n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A073639 Numbers k such that x^k + x + 1 is a primitive polynomial modulo 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A057486 Numbers k such that x^k + x^m + 1 is factorable over GF(2) for all m between 1 and k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A057646 a(n) is the number of trinomials x^n + x^k + 1 that are irreducible over GF(2) for some k with n > k > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A344142 Lexicographically first irreducible polynomial over GF(2) of degree n with the lowest possible number of terms, evaluated at X = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A272486 Numbers n > 1 such that the polynomial x^(n-k)*(x+1)^k+1 over GF(2) is reducible for all k from 1 to n-1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

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.