A267918 Numbers n such that x^(n-5)*(x+1)^5+1 is irreducible in F2[x].

6, 9, 12, 14, 17, 23, 44, 47, 63, 84, 129, 236, 278, 279, 297, 647, 726, 737, 2574, 4233, 8207, 16046, 21983, 23999, 24596, 24849, 84929

Offset: 1

Author

Luis H. Gallardo, May 01 2016

Keywords

• nonn
• more

Comments

Putting M(n,a) = x^(n-a)*(x+1)^n+1 in F2[x], a "Mersenne binary polynomial" and S(n,a) = x^n +(x+1)^a in F2[x], we see that the n's in the sequence are also the n's where S(n,5) is irreducible.

Irreducible Mersenne binary polynomials appear as factors of the only eleven known (see Canaday's paper) nontrivial even perfect polynomials over F2, i.e., polynomials A in F2[x], divisible by x*(x+1), that are fixed points of the sum of divisors function sigma. In other words, we also have sigma(A)=A, where sigma(A) is the sum in F2[x] of all divisors of A (including 1 and A). Trivial even perfect polynomials are the M(2^(n+1)-2,2^n-1)+1 = x^(2^n-1)*(x+1)^(2^n-1).

Next term > 10^5. - Joerg Arndt, May 01 2016

Examples

			For n=6, x^(6-5)*(x+1)^5+1 = x^6 + x^5 + x^2 + x + 1 is irreducible in F_2[x].
		

Links

• E. F. Canaday, The sum of the divisors of a polynomial, Duke Math. J. 8, (1941), 721-737.

Programs

PARI

for(n=5,10^5, if(polisirreducible(Mod(1,2)*(x^(n-5)*(x+1)^5+1)),print1(n,", "))); \\ Joerg Arndt, May 01 2016

Sage

P. = GF(2)[]
for n in range(6, 10^5):
    if (x^(n-5)*(1+x)^5+1).is_irreducible():
        print(n)
# Joerg Arndt, May 01 2016

Extensions

Terms a(12) and beyond from Joerg Arndt, May 01 2016