Number Theory: Are there infinitely many primes of the form k^2 + 3k + 2?

[BigNate]

Hello,

I am trying to answer the following question:

Question: Are there infinitely many primes of the form k^2 + 3k + 2?

I know there are inifinitely many primes of the form 3k + 2. I just cannot figure out how to address the k^2. Can someone please answer this question and explain the logic behind it? I am stumped.

 

[Deleted member 4993]

Hello,

I am trying to answer the following question:

Question: Are there infinitely many primes of the form k^2 + 3k + 2?

I know there are inifinitely many primes of the form 3k + 2. I just cannot figure out how to address the k^2. Can someone please answer this question and explain the logic behind it? I am stumped.

Can you factorize the given quadratic expression?

 

[BigNate]

Can you factorize the given quadratic expression?

So If I view k^2 + 3k + 2 as (k + 2)(k+1)...

...can you please help me a little more? :cool:

My thought is that we need to use the fact that a product of two primes is never prime. There are infinitely many primes of the form k + 2 and k + 1, but the factorization shows us these will not be prime once multiplied. Is this the correct logic or am I completely off?

 

[ksdhart2]

So If I view k^2 + 3k + 2 as (k + 2)(k+1)...

...can you please help me a little more? :cool:

My thought is that we need to use the fact that a product of two primes is never prime. There are infinitely many primes of the form k + 2 and k + 1, but the factorization shows us these will not be prime once multiplied. Is this the correct logic or am I completely off?

Where to go next depends entirely on if k must be an integer or not. If k has to be an integer, then your thinking is on the right track, but I'd instead investigate the parity (i.e. evenness or oddness) of k + 1 and k + 2. Then, what do you know about the product of two odd numbers? Of two even numbers? Of an odd and an even number? How does that relate to the primes?

On the other hand, if k can be any real number, then you might see if you can solve (k + 1)(k + 2) = 3, or try solving (k + 1)(k + 2) = 5. What does that suggest? Because the problem asks are there infinitely many primes of that form, you'll likely want to attempt a proof by induction at some point.

 

[Deleted member 4993]

So If I view k^2 + 3k + 2 as (k + 2)(k+1)...

...can you please help me a little more? :cool:

My thought is that we need to use the fact that a product of two primes is never prime. There are infinitely many primes of the form k + 2 and k + 1, but the factorization shows us these will not be prime once multiplied. Is this the correct logic or am I completely off?

So:

If k is an integer the number of primes of the form "k^2 + 3k + 2" = 0

So the answer to the question:

Are there infinitely many primes of the form k^2 + 3k + 2?

is a resounding NO.