Simultaneous primality and Dirichlet's Theorem on Arthm. Progressions?

[tuttlerice]

Hi. I'm a music theorist writing a music-related paper that is math heavy and I'm a little in over my head.

I know that according to Dirichlet's theorem on arithmetic progressions, there are infinitely many primes in the form ax+b when a and b are coprime. What I am wondering is if there is a theorem that says given ax+b and cx+d where a,b are coprime and c,d are coprime, ax+b and cx+d are simultaneously prime infinitely often?

The best I could come up with trying to prove it myself is this, and I don't claim this is a valid proof:

ax+b and cx+d are both prime infinitely often if it is the case that ax+b+cx+d equals some p+q, where p, q are prime, infinitely often. For the purposes of my paper I can also stipulate that a and c are both even and b and d are both odd.

ax+b+cx+d = x(a+c)+b+d = x(a+c)-1 + b+d+1.

The expression x(a+c)-1 yields primes infinitely often per Dirichlet because a+c is coprime with 1 (as all numbers are coprime with 1). The expression b+d+1 yields primes infinitely often because b+d is even, and even numbers are 1 less than a prime infinitely often. Therefore, ax+b+cx+d = primes p + q infinitely often.

At least that's my train of thought. But I would dearly love for there to be an existing theorem that covers this. And please forgive me if my own "proof" is faulty and naive--- I'm just a music theorist, not a math professor!Robert Gross

 

[tuttlerice]

I also apologize if this is the wrong forum. I thought this was the general number theory forum.

 

[bpet]

If you get the proof right, you will have also proved the twin primes conjecture as a special case. Best of luck!

 

[tuttlerice]

Wow, you're right!

 

[blue_raver22]

Hello Robert,

Thank you for your question. I can provide some insights on this topic. First, let me clarify that there is no known theorem that specifically addresses the simultaneous primality of two arithmetic progressions. However, there are some theorems and conjectures that can give us some insights on this topic.

One such theorem is the Bunyakovsky conjecture, which states that if f(x) is a polynomial with integer coefficients and f(x) takes on infinitely many prime values, then f(x) is irreducible. This means that if we can show that ax+b and cx+d are both irreducible polynomials, then they will take on infinitely many prime values and therefore be simultaneously prime infinitely often.

Another approach is to look at the Bateman-Horn conjecture, which is a generalization of Dirichlet's theorem. It states that for any fixed k, there are infinitely many primes of the form ax+b where (a,k)=1. This means that if we can show that ax+b and cx+d are both in the form of a prime number multiplied by a fixed number, then they will be prime infinitely often.

In conclusion, while there is no specific theorem that addresses the simultaneous primality of two arithmetic progressions, there are some conjectures and theorems that can give us some insights and potentially lead to a proof. I hope this helps in your research. Good luck with your paper!