Re: proove that n^4 + n is never prime

From: Adam (adam_at_bonkers.reg)
Date: 06/10/04 

Date: Thu, 10 Jun 2004 09:50:30 GMT

"DanKage" <eba012@webmail.uib.no> wrote in message
news:nhagc0d64mecpn6edvob9jfe7lnv95kr48@4ax.com...
> Sorry, the statement is that n^4 + 4 is never prime (NOT n^4 + n)
>
>

Proof.
We prove that the statement is not prime by showing that when n is even or
odd, the statement is not prime, and hence it is not prime for all n >= 2.
Let n be an even integer and n >= 2. Then n = 2k for some k, where k is an
integer. n^4 + 4 = (2k)^4 + 4 = 2^4*k^4 + 4 = 4(4k^4 + 1), which is not
prime since it is a multiple of 4. Now let n be an odd integer and n >= 2.
Then n = 2k + 1 for some k, where k is an integer. n^4 + 4 = (2k + 1)^4 + 4
= 4^2(k/2 + 1/2)^4 + 4 = 4[4(k/2 + 1/2)^4 + 1], which is not prime since it
is a multiple of 4. Since n^4 + 4 is not prime if n is even or odd, n^4 + 4
is not prime for all n >= 2. QED.

Since n is an integer it is either even or odd. One can then prove that the
statement is not prime by showing the separate cases are not prime, which
taken together mean that the statement is not prime for all n.

I hope this helps, and good luck!, Adam.

Relevant Pages

  • Re: 3^m + 1 = 10^n
    ... Now for n odd numbers Lhs could be rewritten as: ... once at Rhs we have: ... add: showing instead of ...
    (sci.math)
  • Re: GUI primer, dos and donts
    ... > the problem data. ... I also use light red for showing where ... > data seems odd but will be accepted so the user should ... glyph displayed near control where value is invalid and this control focused ...
    (borland.public.delphi.thirdpartytools.general)
  • Re: Datetime error was Re: Thank you Sysinternals. I can now kill D2006 remotely
    ... > Odd. ... My PC clock is fine, and the control that shows the time that will be ... > assigned to the message is showing the right time, ...
    (borland.public.delphi.non-technical)
  • Re: Very strange goings on at the BBC
    ... recently succesfully challenged the showing of an "inconvenient Truth" to ... odd for the BBC as they have always taken a very subjective emotional ...
    (uk.sci.weather)
  • Re: C++...and Im stumped!
    ... insulting me if you say my code looks odd or confusing, ... About that VB showing, ... There are so many times I get an ambiguous compiler error. ...
    (alt.comp.lang.learn.c-cpp)
Quantcast