Problem 1040

[Richard Guy]

Yes, for  n = 3  it's the classical Markoff
equation, which yields the binary (i.e. trivalent)
tree of triples (D12 ib UPINT).  Presumably
for larger  n you get an {n choose 2}-valent
tree and lots of solutions.  I may not have
found the smallest, or the smallest greater
than 1040.

On Wed, 12 Oct 2005, Joshua Zucker wrote:

> I notice that the n=3 problem is discussed in UPINT, but I don't have
> a copy of UPINT (sorry!).
>
> Is the unsolved part whether or not all solutions (x,y,z) come from an
> earlier solution (y,z,w) where w < x?  This SEEMS to be true -- and I
> checked a bunch of small solutions -- but I certainly haven't proven
> it.
>
> I can see from your "consecutive pair" argument that you're doing that
> same technique.  But you can also generate other solutions: for
> instance, if you put in 1 1 1 1 6 41, you get 1721.  So there's also
> the sequence of largest numbers of a set of 7, that includes stuff
> like that 1721 made by taking any set of previous things from this
> list -- I think deleting any one number from the list always leads to
> a quadratic and thus leads to two possible replacements, the thing you
> deleted and some other number -- what I wonder about is whether there
> are other "primitive" solutions besides 1 1 1  (or 1 1 1 1 1 1 1) in
> this procedure.
>
> Thanks!
>
> --Joshua Zucker
>
>
> On 10/12/05, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
>> 1, 1, 1, 1, 1  and any consecutive pair from
>>
>> 1, 6, 41, 281, 1926, 13201, 90481, 620166, 4250681, ...
>>
>> This is A049685 in OEIS, though this doesn't
>> mention the problem.
>>
>> This works for any value of  7.  E.g. with  31
>> we get
>>
>> 1, 30, 929, 28769, 890910, 27589441, 854381761, ...
>> but this is not (yet) in OEIS.     R.
>