

A057463


Numbers n such that x^n + x^4 + 1 is irreducible over GF(2).


2



1, 3, 7, 9, 15, 39, 57, 81, 105, 1239, 5569, 9457, 11095, 11631, 12327, 37633, 63247
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OFFSET

1,2


COMMENTS

a(18) is greater than 10^5.  Joerg Arndt, Apr 28 2012
All terms are congruent to 1 or 3 (mod 6).  Robert Israel, Sep 05 2016


LINKS

Table of n, a(n) for n=1..17.
Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850


EXAMPLE

6 is not in the sequence since x^6 + x^4 + 1 = (x^3 + x^2 + 1)^2, but 7 is in the sequence since x^7 + x^4 + 1 is irreducible. (Trial division by x + 1, x^2 + x + 1, x^3 + x^2 + 1, and x^3 + x + 1)  Michael B. Porter, Sep 06 2016


MAPLE

for m from 1 to 200 do if(Irreduc(x^m + x^4 + 1) mod 2) then printf("%d, ", m):fi:od: # Nathaniel Johnston, Apr 19 2011


MATHEMATICA

Do[ If[ ToString[ Factor[ x^n + x^4 + 1, Modulus > 2 ] ] == ToString[ x^n + x^4 + 1 ], Print[ n ] ], {n, 0, 750} ]


PROG

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


CROSSREFS

Cf. A002475.
Sequence in context: A304850 A306740 A320022 * A287124 A118258 A117583
Adjacent sequences: A057460 A057461 A057462 * A057464 A057465 A057466


KEYWORD

nonn,more


AUTHOR

Robert G. Wilson v, Sep 27 2000


EXTENSIONS

a(10)a(15) from Nathaniel Johnston, Apr 19 2011
a(16)a(17) from Joerg Arndt, Apr 28 2012


STATUS

approved



