Proving conjectures: if n prime, n^2+n+1; p^2+q^2 not div by

[Kebex]

I need some help proving several questions. I got really stuck at these and don't know where to begin. Other than that, these type of questions seem to come out often in my tests so I really need some help. Here we go;

1. n is a prime => n^2 + n + 1

2. The interior angle of a regular polygon with n sides is (2n-4)/n right angles.

3. Given that p and q are odd numbers p^2 + q^2 cannot be divisible by 4.

Thanks in advance. Explanation of steps would be nice but i can learn from step by step solution if needed.

 

[Denis]

Re: Proving conjectures

3. Given that p and q are odd numbers p^2 + q^2 cannot be divisible by 4.

I'll take this one; all you get is a hint!

Let p = 2a+1 and q = 2b+1 (since they're both odd)

(2a+1)^2 + (2b+1)^2 = 4k where k is any integer
4a^2 + 4a + 1 + 4b^2 + 4b + 1 = 4k

OK??

 

[daon]

Re: Proving conjectures

I'd like to add that #1 and #2 don't make sense.
1. n^2 + n + 1 is not a statement.
2. The only regular polygon that has a right angle is the square.

3. Given that p and q are odd numbers p^2 + q^2 cannot be divisible by 4.

I'll take this one; all you get is a hint!

Let p = 2a+1 and q = 2b+1 (since they're both odd)

(2a+1)^2 + (2b+1)^2 = 4k where k is any integer
4a^2 + 4a + 1 + 4b^2 + 4b + 1 = 4k

OK??

Strong hint, haha.

 

[PAULK]

Re: Proving conjectures

Greetings,

I need some help proving several questions. I got really stuck at these and don't know where to begin. Other than that, these type of questions seem to come out often in my tests so I really need some help. Here we go;

1. n is a prime => n^2 + n + 1

This is meaningless. You didn't say anything.

2. The interior angle of a regular polygon with n sides is (2n-4)/n right angles.
Any standard geometry text will have this.

3. Given that p and q are odd numbers p^2 + q^2 cannot be divisible by 4.
I think this has already been done on this site. Check some of the past answers.


Thanks in advance. Explanation of steps would be nice but i can learn from step by step solution if needed.

 

[Denis]

Re: Proving conjectures

I'd like to add that #1 and #2 don't make sense.
1. n^2 + n + 1 is not a statement.
2. The only regular polygon that has a right angle is the square.

How dare you accuse some poor math teacher of not making sense ? :shock: :shock:

Seriously, many make cents by not making sense :roll:

 

[Kebex]

Oh, sorry

1. If n is a prime => n^2 +n+ q is a prime.

 

[Kebex]

Re: Proving conjectures

3. Given that p and q are odd numbers p^2 + q^2 cannot be divisible by 4.

I'll take this one; all you get is a hint!

Let p = 2a+1 and q = 2b+1 (since they're both odd)

(2a+1)^2 + (2b+1)^2 = 4k where k is any integer
4a^2 + 4a + 1 + 4b^2 + 4b + 1 = 4k

OK??

Why is it 4k?

How can I prove that 4a^2 + 4a + 1 + 4b^2 + 4b + 1 = 4k cannot be divisible by 4 without using examples?

 

[Kebex]

Re: Proving conjectures

For number 2,

It means that a regular polygon with n sides has (2n-4)/n angles that are 90 degrees.

Basically, Regular polygon with n sides => (2n-4)/n number of right angles.

So (2n-4)/n represents the number of right angles in a regular polygon with n sides.

Above are basically interpretations of the question. What i'm suppose prove or disprove is that, a regular polygon with n sides => (2n-4)/n number of right angles.

However,i already know that the only one that has n sides equal to n right angles are squares and rectangles.

 

[mmm4444bot]

Re: Proving conjectures

... What i'm suppose prove or disprove is ...

Have you heard of the concept of disproving a statement by contradiction?

It's a fairly simple concept; if you can find a single case where the statement fails to hold true, then you state it.

Thus, you have disproved the statement.

 

[Deleted member 4993]

Oh, sorry

1. If n is a prime => n^2 +n+ q is a prime.

Where did the 'q' come from -