## Re: Lehmer sequence puzzle

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• ... With Q = 2 and R = 7 we get precisely 3 units, at n = 3, 5, 13. I assume that this is Mike s discovery, but as he is being cagey, I cannot be sure. David
Message 1 of 33 , May 7, 2009

> Discovery:
> Mike Oakes has found a (Q,R) integer pair such that with
> x = sqrt(R)/2 + sqrt(R-4*Q)/2
> y = sqrt(R)/2 - sqrt(R-4*Q)/2
> (x^n+y^n)/(x+y) is a unit precisely 3 times for n > 1.
>
> Questions:
> Can anyone improve on Mike's tally of 3,
> in a situation where the number of units is finite?
> If not, can anyone prove that 3 is the maximum,
> in a situation where the number of units is finite?

With Q = 2 and R = 7 we get precisely 3 units, at
n = 3, 5, 13.

I assume that this is Mike's discovery,
but as he is being cagey, I cannot be sure.

David
• ... Yes, Mike, that last form is by far the neatest, if one removes the unnecessary lucas , which should be well understood, when one speaks of U or V . 1)
Message 33 of 33 , May 10, 2009
<mikeoakes2@...> wrote [with unfailing courtesy]:

> lucasU(338,25,11584)-5*lucasU(338,25,11583)
> I guess Chris would expect the last of these as being the most
> "canonical"? And I concur, it being also the shortest.

Yes, Mike, that last form is by far the neatest,
if one removes the unnecessary "lucas", which should
be well understood, when one speaks of "U" or "V".

1) François Édouard Anatole Lucas defined the integer sequence
U(P,Q,n) = P*U(P,Q,n-1) - Q*U(P,Q,n-2),
with U(P,Q,0) = 0 and U(P,Q,1) = 1,
in terms of elementary arithmetic.

2) I like to imagine that Derrick Norman Lehmer (1867-1938) saw that
U(P,Q^2,2*k+1) = U(P,Q^2,k+1)^2 - (Q*U(P,Q^2,n))^2
and told his son: "Go look at the factors for your Ph.D."

3) Certainly, Derrick Henry Lehmer (1905-1991) did study
U(P,Q^2,k+1) - Q*U(P,Q^2,k)
being very well aware of the super-Lucasian extension to powers
of algebraic numbers of degree 4, in the wider complex plane.

4) Mike Oakes has found that the Lehmer number
U(P,Q^2,k+1) - Q*U(P,Q^2,k)
is probably prime, when P = 338, Q = 5, k = 11583.

5) David Broadhurst is able to characterize this
circumstance without using a dirty 4-letter word :-)

Thank ye, kindly, Sir, for this interesting thread!

David, pp SSSR
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