Conjecture no.25
There is more values of primes that make this set of finite polynomials( a1x^n+b1, a2x^n +b2, a3x^n+b3, …anx^n+bn ) generate more primes compared to another set of finite polynomials with the same amount of elements (a1x^n+c1, a2x^n +c2, a3x^n+c3, … anx^n+cn ) where (c1, c2 , c3, cn)< (b1, b2 , b3, bn). However, we must state some some necessary conditions for this in order for this to be true. The polynomials considered here must have positive integral coefficients and must not be factorable over the integers. The degrees of the polynomials considered here must be greater than zero. The pairwise coprimality of each element of the set of polynomial progression is necessary e.g GCD(a1,b1)=1, GCD(a2,b2)=1, … GCD(an,bn)=1(For the first set of polynomial progressions) The second set must also be pairwise coprime in a likewise manner. Here the italics indicate that the letters and numbers are subscripts. I also wonder which set produces more regular primes versus irregular primes i.e does (b1, b2 , b3, bn) produce more irregular primes than (c1, c2 , c3, cn) assuming that there is infinitely many regular primes implictly and also assuming (b1, b2 , b3, bn) > (c1, c2 , c3, cn) with the same conditions outlined above. Is the conjecture over the larger domain of the Gaussian Primes with also generalizations of regular and irregular primes?
As always I welcome the readers to prove or disprove this conjecture or to extend this conjecture further if possible. Some sources for my conjecture will be displayed below.
https://en.wikipedia.org/wiki/Dickson%27s_conjecture
https://en.wikipedia.org/wiki/Bunyakovsky_conjecture
https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H
https://en.wikipedia.org/wiki/Chebyshev%27s_bias
http://mathworld.wolfram.com/ChebyshevBias.html
http://mathworld.wolfram.com/IrregularPrime.html
http://mathworld.wolfram.com/RegularPrime.html
https://en.wikipedia.org/wiki/Regular_prime
https://oeis.org/A007703
https://en.m.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes
Thank you,
Nunghead
Unfortunately, this conjecture fails due to the modular prime counting function giving examples of linear progressions such that both are equinumerous. However, it is still not known whether there are infinitely many such pairs or tuples such that both are equinumerous. In light of this, I ask whether conjecture no. 25 has only finitely many counterexamples.
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Link for info on modular prime counting function https://mathworld.wolfram.com/ModularPrimeCountingFunction.html
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