> > For integer Q > 0 and integer R

Just exclude oo explicitly... (What is the largest finite value...).

> > with both R and R-4*Q non vanishing,

> > let u(R,Q) be the number of integers n > 0 for which

> > (x^n+y^n)/(x+y) is a unit, with

> > x = sqrt(R)/2 + sqrt(R-4*Q)/2,

> > y = sqrt(R)/2 - sqrt(R-4*Q)/2.

> > What is the maximum value of u(R,Q) when R and Q

> > range over all the integers allowed above?

> >

> > The solution remains very simple: there is no such maximum!

> I posed the puzzle because I have found, experimentally, examples where u(R,Q) is 1, 2 and even (a great surprise to me!) 3.

>

> I want to know if there are any examples >3 (and <oo).

>

> Help me, please, to both formulate the puzzle securely and solve it!

An hint through an experimental approach:

u(R,Q,N=9)={ my(x=sqrt(R)/2+sqrt(R-4*Q)/2,y=sqrt(R)/2-sqrt(R-4*Q)/2);sum(n=1,N,round((x^n+y^n)/(x+y))==1)};

? matrix(20,20,R,Q,u(R,Q))

%415 =

[3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[9 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1]

[2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1]

[1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1]

The numbers in column 1, rows 1..4, grow if you give a larger 3rd argument (as would (Q,R)=(0,1)), but not the others.

Maximilian, per proxy SEAPS

(Society for Experimental Approaches to Puzzle Solving)- --- In primenumbers@yahoogroups.com, "Mike Oakes"

<mikeoakes2@...> wrote [with unfailing courtesy]:

> lucasU(338,25,11584)-5*lucasU(338,25,11583)

Yes, Mike, that last form is by far the neatest,

> I guess Chris would expect the last of these as being the most

> "canonical"? And I concur, it being also the shortest.

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