# Proving the Oddity of n^2+n+1 Directly from Definition

• kuahji
In summary, to prove that n^2+n+1 is always odd for all integers n, we can break it down into cases where n is either even or odd. In both cases, we can show that n^2+n+1 is equal to 2 times some integer plus 1, which is always an odd number. This proof directly uses the definition of even and odd numbers and does not rely on any previously known facts.
kuahji
For all integers n; n^2+n+1 is odd.

Prove the statement directly from definition of the terms & do not use any previously known facts.

My main problem is if n^2+n+1=n(n+1)+1 then can I say n is even & n+1 is odd or would that imply that I'm using previously known facts. In the text there is a section on parity, but I'd hate to have to have a proof inside of a proof. Then even worse, would I then have to go through & prove what even & odd are? Not really sure what "directly from the definition of the terms" implies. Any ideas? Its obvious the professor would know best what she meant but there is no way to contact her atm.

Well what they DON'T want you to do is say "well if n is odd then n^2 is odd, odd number plus an even number(the odd number n and then +1 is even)is always an odd number

if n is even then n^2 is even, even + odd = odd again(n+1 would be odd in the case of even n) That's what they DON'T want to see

You knew before hand as a fact that even * even = even, odd * odd = odd, and even+odd=odd. You would have to prove THOSE if you wanted to use 'em. And you could!

Obviously n is either even or odd right? In each case what is the parity of n(n+1), and hence what is the parity of n^2+n+1?

blochwave said:
You knew before hand as a fact that even * even = even, odd * odd = odd, and even+odd=odd. You would have to prove THOSE if you wanted to use 'em. And you could!

What I did was break it down by cases. But first I said consecutive integers must be even & odd or odd & even, by parity. Then Case 1 was n is odd & hence n+1 is even. Case two n is even & n+1 two is odd (these were drawn out showing the algebraic steps). Then I showed how an odd times an even always equaled an even, 2(integer). But I just got a big x on my paper that said refer to directions, not proven directly from definition of terms & the word parity was underlined.

So does that mean I would have to prove consecutive integers are even/odd since I set it up to n(n+1)+1? Sorry for the confusion, as I'm a bit confused as well.

Or, to be very precise:

n is either even or odd.

case 1: If n is even, n= 2m for some integer m. Then n2+ n+ 1= (2m)2+(2m)+ 1= 4m2+ 2m+ 1= 2(2m2+ m)+ 1, an odd number.

case 2: If n is odd, n= 2m+1 for some integer m. Then n2+ n+ 1= (2m+1)2+ (2m+1)+ 1= 4m2+ 2m+ 1+ 2m+ 1+ 1= 4m2+ 4m+ 3= 2(2m2+ 2m+ 1)+ 1, again an odd number.

## 1. What is the definition of proving the oddity of n^2+n+1 directly?

The definition of proving the oddity of n^2+n+1 directly is using mathematical reasoning and logical steps to demonstrate that the expression n^2+n+1 is always an odd number, for any integer value of n.

## 2. Why is proving the oddity of n^2+n+1 important in mathematics?

Proving the oddity of n^2+n+1 is important in mathematics because it helps to establish the fundamental properties and rules of numbers. It also demonstrates the power and usefulness of mathematical proofs in solving complex problems.

## 3. What are the steps involved in proving the oddity of n^2+n+1 directly?

The steps involved in proving the oddity of n^2+n+1 directly include setting the expression equal to 2k+1 (where k is an integer), simplifying the equation, and using algebraic manipulation and properties to show that the expression will always result in an odd number.

## 4. Can the oddity of n^2+n+1 be proven indirectly?

Yes, the oddity of n^2+n+1 can also be proven indirectly by assuming the expression is even and then reaching a contradiction. This method is known as a proof by contradiction.

## 5. What are some real-world applications of proving the oddity of n^2+n+1 directly?

Some real-world applications of proving the oddity of n^2+n+1 directly include cryptography, computer programming, and engineering. These fields often use mathematical proofs to ensure the accuracy and security of their algorithms and designs.

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