tuttlerice
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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
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
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