Exists a binary primitive pentanomial of degree $n$, for every $n\ge5$?

Question

Or, in other words, prove (or disprove) this conjecture:

$\forall n\ge5,\exists(i,j,k),n>i>j>k>0,\text{ such that}$
$\;x^n+x^i+x^j+x^k+1\text{ is a primitive polynomial in }GF(2)$.

Also: can we bound $i$ as a function of $n$?

See A132451 for small examples.

Note: the question is tagged irreducible-polynomial because primitive polynomials are a subclass of that.

Answer 1

I answered this question on another internet math forum a few years ago:

For $n>4$ exists a primitive pentanomial of degree $n$ with coef. in ${\bf Z}_2$
Posted: Jul 28, 2010 8:01 PM

In article <[email protected]>, Francois Grieu wrote:

Has this assertion been proved?

For any $n>4$ there exists a primitive polynomial of degree $n$, with coefficients in ${\bf Z}_2$, having exactly 5 non-zero terms.

I think this is still open. It was stated by Solomon Golomb as a conjecture in his paper, Periodic binary sequences: solved and unsolved problems, in 2007.

For a few examples: https://oeis.org/A132451

For a few more examples (everything up to n = 400, in fact), http://www.jjj.de/mathdata/pentanomial-primpoly.txt