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- Jun 19, 2020
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Recently I've been looking into Dirichlet's Theorem of Arithmetic Progressions and decided to run a test of my own. Basically I wanted to see if- similar to how primes of the form Ax+B are evenly divided given different coprime values of B for some constant A- are seniprimes ALSO evenly divided among quadratics of some form given some contents and a varying value?
The first step was to determine what the form of this quadratic would be, as clearly an unfactorable quadratic couldn't result in a semiprime for all integer variables plugged in. The format I chose to use was A^2x^2 + 2ABx + (C^2 - B^2), or, when factored out, (Ax + (B - C)) * (Ax + (B + C)). This means the quadratic would result in a semiptime whenever there occurs a pair of primes with a difference of 2C and an average of Ax + B.
Now the first thing you'll probably say is "Well, obviously if the full proof of Dirichlet's Theorem could apply to this, then you would have a proof for the Twin Primes conjecture and Polignac's conjecture, as you'd have shown there are infinitely many primes of any gap 2C!", and you'd be right to say that, but that's not the part I'm looking into right now. Right now I'm more interested in knowing if semiprimes occur equally often on versions of this quadratic given constants for A and C, and varying B.
Now, I don't know anything about complex analysis of L functions, so a proof is way beyond me, but what isn't beyond me is coding in python. So I made a simple program which counts how many semiprimes occur given these parameters, and my suspicions were only reinforced. Given A = 10 and C = 1, there are 3 values for B which can be used. B = 2, 4, and 10 (B <= A). On each of these 3 progressions, each one counted almost exactly the same number of semiprimes up to x = 100,000, or about 2700 each if I recall.
This could've been a fun coincidence, but the same pattern happened for many other inputs for A and C. Another interesting thing about the number of valid inputs for B is that it is equal to the number of pairs of positive integers (B-C) and (B+C) where B<=A, such that both integers are coprime with A, seemingly an alternate version of Euler's Totient function given pairs of values instead of single ones.
Now for the actual question portion of this- would anyone know how you would go about proving something like this, assuming it's possible? And if so, what does that proof imply exactly?
The first step was to determine what the form of this quadratic would be, as clearly an unfactorable quadratic couldn't result in a semiprime for all integer variables plugged in. The format I chose to use was A^2x^2 + 2ABx + (C^2 - B^2), or, when factored out, (Ax + (B - C)) * (Ax + (B + C)). This means the quadratic would result in a semiptime whenever there occurs a pair of primes with a difference of 2C and an average of Ax + B.
Now the first thing you'll probably say is "Well, obviously if the full proof of Dirichlet's Theorem could apply to this, then you would have a proof for the Twin Primes conjecture and Polignac's conjecture, as you'd have shown there are infinitely many primes of any gap 2C!", and you'd be right to say that, but that's not the part I'm looking into right now. Right now I'm more interested in knowing if semiprimes occur equally often on versions of this quadratic given constants for A and C, and varying B.
Now, I don't know anything about complex analysis of L functions, so a proof is way beyond me, but what isn't beyond me is coding in python. So I made a simple program which counts how many semiprimes occur given these parameters, and my suspicions were only reinforced. Given A = 10 and C = 1, there are 3 values for B which can be used. B = 2, 4, and 10 (B <= A). On each of these 3 progressions, each one counted almost exactly the same number of semiprimes up to x = 100,000, or about 2700 each if I recall.
This could've been a fun coincidence, but the same pattern happened for many other inputs for A and C. Another interesting thing about the number of valid inputs for B is that it is equal to the number of pairs of positive integers (B-C) and (B+C) where B<=A, such that both integers are coprime with A, seemingly an alternate version of Euler's Totient function given pairs of values instead of single ones.
Now for the actual question portion of this- would anyone know how you would go about proving something like this, assuming it's possible? And if so, what does that proof imply exactly?