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Re: More on Leroy Quet's prime trees.
Posted:
Dec 15, 2002 9:55 PM
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mathwft@math.canterbury.ac.nz (Bill Taylor) writes:
|> For 15-trees, there are apparently six feasible parity-patterns, and, of
|> course, a great many example trees. I haven't checked that there is at
|> least one for each of the six patterns, (exercise for the reader), but
|> I guess there is, as it is fairly easy to construct examples.
Yes there is at least one of each type, I just checked. It's not hard.
There is another special type that might be fun to search for, and will be
rather easier to program a machine search for, thus maybe giving us some
clue on the matter of whether there are infinitely many overall, or just
(as I suspect) finitely many.
We can also look for "sequential prime trees". These are as before, but
with the further property that all the levels above the bottom have the
integers in sequential order from 1 onwards. It's easy to check there
are none with 2 or 3 levels; and the 4-levellers only *just* miss out too.
The nearest miss is:
1
/ 2 3
/ \ / 4 5 6 7
/ \ / \ / \ / 10 12 11 15 9 13 8 14
totals: 17 19 19 23 19 23 19 25 <----- not quite a prime.
It's not very hard to construct a sequential prime tree with 5 levels;
indeed I expect there are several of them. And I expect they continue
to be just as easy to construct as prime trees in general; up to the point
where the number of levels hits the failure barrier for trees in general,
if there is such a thing.
But as I say, I expect sequential trees are very much easier to search for,
so if anyone cares to try a computer search, it would be fun to see if and
where the search broke down.
Cheers,
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Bill Taylor W.Taylor@math.canterbury.ac.nz
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"There are only 10 kinds of people in the world --
Those who understand binary, and those who don't."
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