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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.

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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

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  • $\begingroup$ Just to make it clear: that was me, about 4 years ago. I asked again because math.SE is perfect for this purpose, and there might have been progress/extra info. $\endgroup$
    – fgrieu
    Commented Jul 7, 2014 at 17:05
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    $\begingroup$ It would not have been a bad idea to include that information in the body of your question. $\endgroup$
    – Gerry Myerson
    Commented Jul 8, 2014 at 0:00
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    $\begingroup$ I have done some searching through Math Reviews, and haven't found anything more recent. You might try asking on MathOverflow (but please be sure to link the two questions, at both sites). $\endgroup$
    – Gerry Myerson
    Commented Jul 8, 2014 at 0:09

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