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Old 2019-10-05, 12:53   #35
Dr Sardonicus
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Feb 2017

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Thanks to R.D. Silverman and jyb for the references.

As it turns out, I've actually had both of Bent's papers in my files for over a decade.

I'd forgotten there were two basic identities. Now that I'm reviewing the papers, I even vaguely recall having plowed through them at the time I originally downloaded them.

There are a few remaining algebraic details about the specific application to generalized Lucas numbers, but I'm sure I can work these out.

BTW, looking at this stuff made me think of a longstanding question: For which squarefree N > 1 does the fundamental unit of the quadratic field Q(sqrt(N)) have norm -1? It is well known, of course, that that N = 2, or N = p, a prime congruent to 1 (mod 4) fills the bill. And, of course, it is necessary that -1 be a quadratic residue of every prime factor of N. But beyond that, I'm not sure what has been found.

Last fiddled with by Dr Sardonicus on 2019-10-05 at 12:56 Reason: Add necessary condition
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