Question on Prime Numbers (1 Viewer)

[Shadowdude]

So I have to prove that q(n) = 11n^2 + 32n is prime for only two integer values of n, and composite for all others.

My working is:

q(n) = n(11n+32)

And then n = +/- 1 or 11n+32 = +/- 1. Solve for n and work out q(n) in the cases.

The case for why n or 11n+32 must be 1 is easy enough, but why do we allow the case for n = -1?

Further, why do we those as the only cases to check?

 

[lolcakes52]

Prime numbers have to be greater than 1. So i think the answer is n=-1 and -3. The sign on n is not important to determine whether q(n) is prime. Just as P(x)=3+x is prime for x=-1 even though -1 is negative.

 

[Shadowdude]

The solutions are n = 1 and n = -3, yes. But the question is "Why is q(n) prime for only two values of n, and composite for all others?"

I know it has something to do with the factorisation being n and 11n+32, but... yeah. I'm being buried in work

 

[jet]

Well a prime number can only be resolved into the factors 1 and itself. Hence, 11n + 32 must be irreducible (think about it: if 11n + 32 were anything but irreducible then q(n) wouldn't be prime) and so should n. Either n will be 1 or 11n + 32 will be 1 for it to work.

 

[Shadowdude]

But if 11n + 32 = 1, then n = -31/11, which isn't an integer input.

In fact, if 11n + 32 = -1, we get n = -3,, which is the other solution to q(n) being prime. So my question then is, why are we testing if 11n + 32 = -1 and if n = -1.

I could just say it works because "it works", but... why?

 

[IamBread]

But if 11n + 32 = 1, then n = -31/11, which isn't an integer input.

In fact, if 11n + 32 = -1, we get n = -3,, which is the other solution to q(n) being prime. So my question then is, why are we testing if 11n + 32 = -1 and if n = -1.

I could just say it works because "it works", but... why?

You just need it to = +-1 so it's prime.

 

[Shadowdude]

But why the case for -1?

 

[IamBread]

But why the case for -1?

Because its a 1, so its prime lol

 

[Shadowdude]

I don't... understand.

So we factorise it to get q(n) = n(11n+32)

n = 1, or 11n+32 = 1 - I get because if q(n) is prime, one of the factors has to be 1.

But I still don't get why we consider the case n = -1 and 11n+32 = -1. (but it's getting late so... my brain might be failing again)

 

[IamBread]

Damn lateness...

So we have q(n) = n(11n+32). Now for q(n) to be prime, either n = +-1 or 11n+32 = +-1. If n = -1, then q(n) is negative, so it's not prime. But if we have 11n+32 = -1, then we have q(n) positive, and prime. So if you sub n = -3, you get q(n) = 3, which is prime.

n can be any integer, so it can be negative. We want either 11n+32 or n to be +-1, so it is prime. Though if one of them is -1, then the other one must be negative to so we end up with a positive answer for q(n).

This is probably a bad explanation, but I am also tired lol

 

[Shadowdude]

Aha, that kinda makes sense.

 

[D94]

I thought of it like this:

Intuitively, we know that 11n + 32 must be prime;

11n + 32 = x
11n = x - 32
n = (x - 32)/11

Now, (x - 32)/11 must also be prime, so the smallest +/- solution for x is -1.

Probably rigorously incorrect somewhere, but like your reasoning, "it works"

 

[IamBread]

I thought of it like this:

Intuitively, we know that 11n + 32 must be prime;

11n + 32 = x
11n = x - 32
n = (x - 32)/11

Now, (x - 32)/11 must also be prime, so the smallest +/- solution for x is -1.

Probably rigorously incorrect somewhere, but like your reasoning, "it works"

I don't think that works, because it isn't necessary that 11n + 32 must be prime. We have q(n) = ab, and for q(n) to be prime, ab = a, or ab = b, ie. a or b = +-1. The way you do it doesn't show that q(n) is only prime for 2 numbers.

 

[seanieg89]

My logic:

-If either n or 11n+32 has a factor d with |d|>1, then 1<|d| is a positive divisor of n(11n+32).

-n(11n+32) is prime => |d|=n(11n+32) => n(11n+32) divides either n or 11n+32. This means that one of the two factors must be +-1.

-Check the few values of n which make one of the factors +-1 to see which ones actually make n(11n+32) prime. (Remember, the first two steps find a NECESSARY condition for our expression to be prime, not a sufficient one.)

 

[D94]

I don't think that works, because it isn't necessary that 11n + 32 must be prime. We have q(n) = ab, and for q(n) to be prime, ab = a, or ab = b, ie. a or b = +-1. The way you do it doesn't show that q(n) is only prime for 2 numbers.

True, I guess I was only solving for one solution of n.