Jacob Fugal wrote:

> On 9/1/06, Hans Fugal <fugalh / xmission.com> wrote:
> 
>> Paul Lutus wrote:
>> > "Unhappy numbers have eventually periodic sequences of s(sub)i which 
>> never
>> > reach 1."
>>
>> And what stops it from being a 1x10^50 length period, or any arbitrary
>> length?
> 
> 
> The period cannot be any larger than 1000. The largest period is
> probably quite a bit smaller than this, but this is a upper bound.
> Why?
> 
> Some definitions, first.
> 
> Let's define g(x) as the function that takes a number to it's
> successor. For example, g(42) = 4^2 + 2^2 = 20.
--snip--

Actually, it is possible to show that for any three digit number

g(x) < x.

This holds in any base. The proof:

Let the base be  b > 1, and the number x be
x = u + b * v + b^2 * w,
with 0 <= u, v, w < b, and w > 0.
Then

x - g(x) = u + b * v + b^2 * w - (u^2 + v^2 + w^2)
  = u (1 - u) + v * (b - v) + w * (b^2 - w)
  > u (1 - u) + b^2 - 1
  > (b - 1) (2 - b) + b^2 - 1 = 3 * (b - 1) > 0

Regards,

Michael


-- 
Michael Ulm
R&D Team
ISIS Information Systems Austria
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