Collatz-like conjecture using integer square root [Math]

From: Carl W. on 7 Jul 2010 08:07

On 7 July, 12:47, master1729 <tommy1...(a)gmail.com> wrote:
> googlenews wrote :
>
>
>
> > A quick Google search for some of the numbers below
> > doesn't seem to
> > turn anything up on this, so I thought I'd post it
> > here to see if
> > anyone else has come across it, or finds it
> > interesting.
>
> > Conjecture; the transformation:
> > x -> floor(sqrt(x)) * ( x - floor(sqrt(x))^2 )
> > [for x a non-negative integer, and floor and sqrt
> > rt being "largest
> > integer <= x" and "positive square root of x"
> > respectively, and the
> > asterisk, for the more pedantic, being defined as
> > multiplication]
> > ...will, under repeated iteration a) reach zero or b)
> > become trapped
> > in a finite loop of integers, and c) NEVER diverge to
> > infinity.
>
> > Several things become apparent without much work:
> > i) Starting with x = 0 is a self-terminating case by
> > definition
> > ii) x = [any perfect square] necessarily leads to
> > zero on the next
> > iteration, since the value of the subtraction will be
> > zero
> > iii) x = 8 iterates back to 8 and is thus a loop of
> > one integer.
> > Technically this also applies for zero, but since
> > that is declared as
> > a final state, zero is considered not to be a loop.
> > iv) Every iteration generates a composite number
> > v) The average value of the right hand side is
> > floor(sqrt(x))^2,
> > suggesting an overall drop in value
> > vi) As a direct consequence of (iv), there is a
> > chance, should
> > floor(sqrt(x))^2 and (x-floor(sqrt(x))^2) share
> > common factors, that
> > their product is a perfect square, effectively
> > decreasing the naively
> > calculated average RHS further.
> > vii) None of this necessarily counters the fact that
> > (naively
> > speaking) half the time, the value of the right hand
> > side is greater
> > than x, which could, theoretically at least,
> > contribute to a
> > divergence to infinity.
> > viii) Numbers one less than a perfect square (n^2-1)
> > generate
> > 2*(n-1)^2 as the next iteration, which guarantees at
> > least one further
> > iteration (unless n was 1) since twice a nonzero
> > perfect square is
> > never a square.
>
> > example:
> > Starting with 99 (10^2-1 to make things as
> > interesting, yet hopefully
> > brief, as possible), we have
> > 99
> > -> 9 * 18 = 162
> > -> 12 * 18 = 216
> > -> 14 * 20 = 280
> > -> 16 * 24 = 384
> > -> 19 * 23 = 437 -- odd composites are not impossible
> > of course
> > -> 20 * 37 = 740 -- by this stage all looks hopeless.
> > we seem to be
> > heading to infinity already
> > -> 27 * 11 = 297 -- a surprise. 740 was only 11 more
> > than 27^2 and so
> > the next term is smaller
> > -> 17 * 8 = 136 -- now it seems we may reach zero
> > soon
> > -> 11 * 15 = 165
> > -> 12 * 21 = 252
> > -> 15 * 27 = 405 -- but again we are heading in the
> > wrong direction
> > -> 20 * 5 = 100 -- a product of non squares forming a
> > square...
> > -> 10 * 0 = 0 -- .. and so the next term is
> > necessarily zero, and we
> > have finished.
>
> > Much like the better known Collatz iteration, we seem
> > to have
> > hailstone-like behaviour, rising and falling
> > repeatedly before
> > reaching the goal.
>
> > Research with a computer suggests that approximately
> > 95% of the
> > numbers below 2500000 reach zero in this manner. A
> > further 3% reach 8
> > and become trapped in the loop mentioned in (iii)
> > above.
>
> > The remainder become trapped in other, rarer, loops,
> > of which I
> > suspect there to be an infinite number; A secondary
> > conjecture if you
> > will, call it (d).
>
> > The first few loops, in order of probability of
> > occurrence and
> > starting with smallest term, are:
> > # 8 -> 8 (1 term, included here for completeness)
> > # 1927 -> 3354 -> 5985 -> 4312 -> 5655 -> 2250 ->
> > 1927 (6 terms)
> > # 18469 -> 32940 -> 32399 -> 64082 -> 18469 (4 terms)
> > # 46208 -> 88168 -> 163392 -> 71104 -> 92568 -> 46208
> > (5 terms)
> > # 39852 -> 49949 -> 49060 -> 48399 -> 95922 -> 136269
> > -> 39852 (6
> > terms, and peculiarly has a smaller smallest term
> > than the more
> > frequently encountered loop above)
> > # 1816705 -> 3092712 -> 3776184 -> 1816705 (3 terms)
>
> > Curios:
> > # In a parallel with happy and unhappy/sad numbers
> > (another integer
> > iterative ruleset which sometimes loops), I have been
> > calling those
> > initial numbers that do not reach zero, 'melancholy'
> > numbers.
> > # 881217 is the number less than one million with the
> > longest path to
> > zero, with 220 steps required. The maximum 'hailstone
> > height' on the
> > way is 5461962688.
>
> > The main question:
> > Is there any obvious way of proving (c) and (d) - as
> > an amateur who
> > misses obvious things often, this is not necessarily
> > clear to me - or
> > does this fall into the same category as the Collatz
> > conjecture and
> > there is currently no means to do so?
>
> > Carl
>
> your not the first to notice this intresting iteration.
>
> the master , tommy1729 , has already considered it and posted about it on sci.math a few years ago.
>
> back in the good old times when quasi and galathaea were still around.
>
> hell , i could even state you stole the idea from me , since it also appeared on sci.math , however i will be nice today and give you the benefit of the doubt that you rediscovered this.
>
> i wish you luck , more luck than i got here ...
>
> regards
>
> the master
>
> tommy1729

My apologies to thee, master tommy of the taxicab number. I searched
sci.math with the terms 1927 and 18469 since these would have been
likely to reveal another's work on the same subject, but found
nothing. Perhaps I should have searched 1729 instead.

Carl