## Re: Lehmer sequence puzzle

Expand Messages
• ... I was careless in defining the exact ranges of Q and R. The transformation (R= -R, Q= -Q) leaves LV() invariant; so w.l.o.g. we can require Q 0 (if we
Message 1 of 33 , May 7, 2009
--- In primenumbers@yahoogroups.com, "Maximilian Hasler" <maximilian.hasler@...> wrote:
>
> --- In primenumbers@yahoogroups.com, "Mike Oakes" wrote:
> > LV(R,Q,n) = (x^n+y^n)/(x+y)
> > where x = sqrt(R)/2+sqrt(R-4*Q)/2
> > and y=sqrt(R)/2-sqrt(R-4*Q)/2
> > For given R,Q, define u(R,Q) to be the number of values of n
> > for which LV() is a unit (i.e. +/-1).
> >
> > Puzzle: what is max{R>=1,Q>=1}(u(R,Q))?
>
> R=4, Q=1 => x=sqrt(4)/2+sqrt(4-4)/2 = 1+0 = 1 , y=1-0 = 1
>
> LV(4,1,n) = (1^n+1^n)/(1+1) = 1 for all n.
>
> Thus, u(4,1) = oo = max(u(R,Q); R,Q >= 1)
>
> Maximilian, per proxy SVSR
> (subcommittee for vanishing square roots).

I was careless in defining the exact ranges of Q and R.

The transformation (R=>-R, Q=>-Q) leaves LV() invariant;
so w.l.o.g. we can require Q>0 (if we exclude Q=0 as a degenerate case).

R=0 and R=4*Q are also clearly degenerate cases, to be excluded.

We should then let R range over all integer values (not just R>=1), in fact.

Note: if we do this, there is no need to look at the superficially distinct Lehmer sequence
LU(R,Q,n)=(x^n-y^n)/(x-y)
as this is identical to the sequence LV(4*Q-R,Q,n).
This is in interesting contrast to the Lucas-sequence case.

Mike
• ... 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
Your message has been successfully submitted and would be delivered to recipients shortly.