--------------000001000702080107000506 Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Jacob Fugal wrote: > WARNING: Just more math geeking ahead. If you don't care for this > section of the thread, just skim on past... :) > > On 9/4/06, Michael Ulm <michael.ulm / isis-papyrus.com> wrote: > >> Let the base be b > 1, and the number x be >> x + b * v + b^2 * w, >> with 0 < , v, w < b, and w > 0. >> Then >> >> x - g(x) + b * v + b^2 * w - (u^2 + v^2 + w^2) >> (1 - u) + v * (b - v) + w * (b^2 - w) >> > u (1 - u) + b^2 - 1 >> > (b - 1) (2 - b) + b^2 - 1 * (b - 1) > 0 > > > A nitpick, it should be: > > u (1 - u) + v * (b - v) + w * (b^2 - w) > (1 - u) + b^2 - 1 > > rather than a strict less than. Doesn't affect the outcome of the > proof, since the following inequality is still correct. In the case > where v and w --snip more explanations-- You are right, thanks for the correction and explanations. Some other simple facts: If x is a two-digit number in base b then g(x) < 2 * b^2, so if g(x) is a three-digit number in base b, the most significant digit is always one. This also means that all cycles in base b consist of numbers less than 2 * b^2. No odd number b is a happy base, because the number x b + 1)**2 / 2 is a fixed point of g; i.e. g(x) . The only happy bases < 700 are 2 and 4. To get more on topic for this list, I have attached a ruby script that computes all cycles for any given base. Regards, Michael -- Michael Ulm R&D Team ISIS Information Systems Austria tel: +43 2236 27551-219, fax: +43 2236 21081 e-mail: michael.ulm / isis-papyrus.com Visit our Website: www.isis-papyrus.com --------------------------------------------------------------- This e-mail is only intended for the recipient and not legally binding. Unauthorised use, publication, reproduction or disclosure of the content of this e-mail is not permitted. This email has been checked for known viruses, but ISIS accepts no responsibility for malicious or inappropriate content. --------------------------------------------------------------- --------------000001000702080107000506 Content-Type: application/x-ruby; name apnr.rb" Content-Transfer-Encoding: base64 Content-Disposition: inline; filename apnr.rb" IyBoYXBuci5yYgojCiMgc29tZSBmdW5jdGlvbnMgZXRjLiBjb25jZXJuaW5nIGhhcHB5IG51 bWJlcnMKCgpjbGFzcyBIYXBweU51bQogIGRlZiBpbml0aWFsaXplKGJhc2UpCiAgICBAYmFz ZSA9IGJhc2UKICAgIEBiYXNlX3NxdWFyZSA9IGJhc2UgKiBiYXNlCiAgZW5kCgogIGRlZiBt YXBwZXIodmFsKQogICAgcmVzdWx0ID0gMAogICAgd2hpbGUgdmFsID4gMAogICAgICByZXN1 bHQgKz0gKHZhbCAlIEBiYXNlKSAqKiAyCiAgICAgIHZhbCAvPSBAYmFzZQogICAgZW5kCgog ICAgcmVzdWx0CiAgZW5kCgogIGRlZiBjb21wdXRlX2N5Y2xlcwogICAgY3ljbGVzID0gW10K ICAgIHVzZWQgPSBbXQogICAgQGJhc2Vfc3F1YXJlLnRpbWVzIGRvIHx0fAogICAgICB1bmxl c3MgdXNlZFt0XQogICAgICAgIHRyYWNlID0gW10KICAgICAgICBjdXJyZW50ID0gdAogICAg ICAgIHdoaWxlIG5vdCB1c2VkW2N1cnJlbnRdCiAgICAgICAgICB1c2VkW2N1cnJlbnRdID0g MQogICAgICAgICAgdHJhY2UucHVzaChjdXJyZW50KQogICAgICAgICAgY3VycmVudCA9IG1h cHBlcihjdXJyZW50KQogICAgICAgIGVuZAogICAgICAgIGlmIHVzZWRbY3VycmVudF0gPT0g MQogICAgICAgICAgY3ljbGVzLnB1c2godHJhY2VbdHJhY2UuaW5kZXgoY3VycmVudCkgLi4g LTFdKQogICAgICAgIGVuZAogICAgICAgIHRyYWNlLmVhY2gge3x0fCB1c2VkW3RdID0gMn0K ICAgICAgZW5kCiAgICBlbmQKICAgIGN5Y2xlcwogIGVuZAplbmQKCmlmICQwID09IF9fRklM RV9fCiAgaWYgQVJHVi5lbXB0eT8KICAgIDIudXB0bygzMikgZG8gfGJhc2V8CiAgICAgIHB1 dHMgIiN7YmFzZX06IgogICAgICBwIEhhcHB5TnVtLm5ldyhiYXNlKS5jb21wdXRlX2N5Y2xl cwogICAgZW5kCiAgZWxzZQogICAgICBwIEhhcHB5TnVtLm5ldyhBUkdWWzBdLnRvX2kpLmNv bXB1dGVfY3ljbGVzCiAgZW5kCgplbmQK --------------000001000702080107000506--