Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $d$ be a prime number. Is the polynomial $x^d+x+1$ a primitive polynomial? In other words I need the minimal polynomial of $\alpha$ in $F_{2^d}=F_{2}(\alpha)$.

Thank you.

share|improve this question
1  
Not always! $x^5+x+1 = (x^2+x+1)(x^3+x^2+1)$ –  Will Sawin Jun 28 '13 at 17:27
    
If it is primitive for $d=p$, it will not be primitive for any $d\cong p$ modulo $2^p-1$, because then the polynomial will again be a multiple of $x^p+x+1$. The $d$ for which it is primitive are determined by a prime-like sieve, but they are not the primes. They are almost certainly less frequent than the primes. –  Will Sawin Jun 28 '13 at 17:30
    
Such primitive polynomials yield binary Hamming codes. The number of monomials need not be 3 (but odd) in general, e.g., $x^8+x^4+x^3+x^2+1$ is primitive, too (but $x^8+x+1$ not). –  Dietrich Burde Jun 28 '13 at 20:35

1 Answer 1

up vote 6 down vote accepted

See http://oeis.org/A073639 for the list of n such that x^n+x+1 is primitive modulo 2. The sequence begins [2,3,4,6,7,15,22,60,63,127,153,471,532,865,900,1366] You will also find some pertinent references there.

I found this using Maple's function "Primitive(f(x)) mod p" which returns true if f(x) is primitive mod p, and false otherwise.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.