Infinitely many primes of the form p² + nq² -- really?

[timmdeeg]

Hi,

after stumbling about We show that there are infinitely many primes of the form p² + nq² with both p and q prime ... I found that using n=4 the formula gives 205 and 221 with p=3, q=7 and p=5, q=7 respectively.

So, how to understand that?

 

[PeroK]

Hi,

after stumbling about We show that there are infinitely many primes of the form p² + nq² with both p and q prime ... I found that using n=4 the formula gives 205 and 221 with p=3, q=7 and p=5, q=7 respectively.

So, how to understand that?

Does the paper claim that all numbers of that form are prime?

 

[martinbn]

Hi,

after stumbling about We show that there are infinitely many primes of the form p² + nq² with both p and q prime ... I found that using n=4 the formula gives 205 and 221 with p=3, q=7 and p=5, q=7 respectively.

So, how to understand that?

You should understand it the way it is written! There are infinitely many primes of that form, not all numbers of that from are prime.

 

[timmdeeg]

You should understand it the way it is written! There are infinitely many primes of that form, not all numbers of that from are prime.

Ah, yes. Thanks for clarifying!

Wouldn't this imply that this form gives indefinitely non-primes?

 

[PeroK]

Ah, yes. Thanks for clarifying!

Wouldn't this imply that this form gives indefinitely non-primes?

Not necessarily. Although it's easy to prove. Hint: take $n =6$.

PS or $n =4$.

 

[timmdeeg]

Not necessarily. Although it's easy to prove. Hint: take $n =6$.

PS or $n =4$.

Even it's easy to prove, my mathematical abilities aren't sufficient. My guess is that there are infinitely non-primes, because if the formula gives infinitely primes then there are infinitely gaps with non-primes in between.

Would you mind to show the prove?

 

[PeroK]

Even it's easy to prove, my mathematical abilities aren't sufficient. My guess is that there are infinitely non-primes, because if the formula gives infinitely primes then there are infinitely gaps with non-primes in between.

Would you mind to show the prove?

If we take $p=2$ and $n =4$, then the expression is even for every choice of prime $q$.

Likewise , for $p=3$ and $n =6$, the expression is divisible by 3 for every choice of $q$.

 

[timmdeeg]

Thanks!

 

[WWGD]

I think you're misunderstanding that $n$ is a(n) positive Integer variable; $n=1,2,..$.