What do you use for your basis step when its domain is all integers?

Question

Example: *For all integers $ n , 4( n ^2 + n + 1) – 3 n ^2$ is a perfect square what should i use? negative infinity?

I know you can use a direct proof but what if theres an induction question with the same domain?

Answer 1

You need to use induction twice, once going down and once going up. You can choose any (finite) integer as the base step.

Answer 2

$4(n^2+n+1)-3n^2=4n^2+4n+4-3n^2=n^2+4n+4=(n+2)^2$ So it is a square for all integers $n\in Z$

Induction for negative integers: First you can prove the proposition for positive integers $n$ and then replace $n$ by $-n$ in the proposition and again prove that proposition(formed after replacing $n$ by $-n$) for positive integers.

Answer 3

Negative infinity is not a number. Do not try to do arithmetic with it.

If you were going to do a proof by induction (by the way, you should be clearer in your question that this is your intention) then choose any number as your base case but show that induction works in both directions. i.e. if the $k$'th case holds, then both the $(k+1)$'th and $(k-1)$'th cases hold. Do you see how this proves the statement?

Despite this, I would search for a direct proof of this statement.