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June 6th, 2016, 02:04 PM   #11
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Nice! Thanks for also looking into this! I've also started a check of even termed CFEs and they all seem to hold to 4n+1! I though to try this based on the 4n+3 primes comment from https://oeis.org/A002145:

Bernard Frénicle de Bessy discovered that such primes cannot be the hypothenuse of a a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after Paul Curtz, Sep 10 2008

So, I've got an additional conjecture that all even termed CFE primes are of the form 4n+1.
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June 7th, 2016, 09:01 PM   #12
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Interesting, I will check this too (up to 1 billion of course) when I have time (probably this weekend).
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June 12th, 2016, 02:08 PM   #13
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The 1 billion run is complete and to me it appears that all the primes, and only the primes of the form $4n+1$ have an even number of coefficients (except for 2 of course, which has an even number of terms per your definition but is not of the form $4n+1$).

It seems that you have done the research on possible prior knowledge of these properties. Have you thought about writing up your findings? (No big deal if you don't want to).
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June 21st, 2016, 04:55 AM   #14
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I'd still like to understand the false positive cases a bit better before I write anything formal up, as it feel a little incomplete until then. I've also been studying the matrix forms of the CFEs to try and get another angle on this.
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