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A057751 Irreducible trinomials of prime degree for some k: x^p + x^k + 1 is irreducible over GF(2) for at least one k, p>k>0. 1

%I

%S 2,3,5,7,11,17,23,29,31,41,47,71,73,79,89,97,103,113,127,137,151,167,

%T 191,193,199,223,233,239,241,257,263,271,281,313,337,353,359,367,383,

%U 401,409,431,433,439,449,457,463,479,487,503,521,569,577,593,599,601

%N Irreducible trinomials of prime degree for some k: x^p + x^k + 1 is irreducible over GF(2) for at least one k, p>k>0.

%e The prime 79 is included because x^79 + x^9 + 1 is irreducible over GF(2). Only the primes 2 and 3 are irreducible for all ks between 0 and p. So far about one-half of all trinomials of a prime power are irreducible over GF(2) for at least one k between 0 and p.

%t Do[ k=1; While[ ToString[ Factor[ x^Prime[n ] + x^k + 1, Modulus ->2 ] ] != ToString[ x^Prime[n ] + x^k + 1 ] && k < Prime[n ], k++ ]; If[ k != Prime[ n ], Print[ Prime[ n ] ] ], {n, 1, 100} ]

%K nonn

%O 0,1

%A _Robert G. Wilson v_, Oct 30 2000

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Last modified August 25 03:01 EDT 2019. Contains 326318 sequences. (Running on oeis4.)