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 June 6th, 2016, 02:04 PM #11 Newbie   Joined: May 2016 From: Maine Posts: 6 Thanks: 0 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.
 June 7th, 2016, 09:01 PM #12 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 625 Thanks: 87 Math Focus: Electrical Engineering Applications Interesting, I will check this too (up to 1 billion of course) when I have time (probably this weekend).
 June 12th, 2016, 02:08 PM #13 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 625 Thanks: 87 Math Focus: Electrical Engineering Applications 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).
 June 21st, 2016, 04:55 AM #14 Newbie   Joined: May 2016 From: Maine Posts: 6 Thanks: 0 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.

### "middle coefficient" periodic continued fraction

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