Expert: Paul Klarreich - 11/6/2008

Question
Disprove this statement: There exists an integer n such that n^3-n+1 is even.

Answer
Questioner:   Pete
Category:  Calculus
Private:  No
 
Subject:  More Proofs!!
Question:  Disprove this statement: There exists an integer n such that n^3-n+1 is even.
.......................
Hi, Pete,

Write the definitions:

n is even if it can be written  n = 2k, for some integer k.
n is odd if it can be written  n = 2k + 1, for some integer k.

Now apply this to show that no matter what,  n^3 - n + 1 is odd.  You should be able to take it from there.
.................................................
Next time, don't just copy the question.  Tell me what you already tried to do, and what you are studying.  {It said this in my Instructions to Questioners, which include:

(1) Don't mark 'Private'.
(2) PROOFREAD.  If you have obvious errors and mistyping, I will conclude you are not serious about your question.  Then I won't be serious about answering.
(3) First check the archives.[Click Browse Past Answers and search for a topic like yours, such as RELATED RATES.  You might find your problem there and you won't have to wait for me.]
(4) Next, check the archives to see HOW to write mathematical expressions so I can understand your question.
(5) Include the WHOLE question, with the instructions.
(6) Tell me what you already did, and what you are studying.  
(7) Don't use special jargon from engineering or business applications. I won't understand.