# [seqfan] Distribution of the Prime Numbers in a Polynomial

Charles Kusniec ck at centertap.com.br
Tue May 16 14:11:13 CEST 2017

```Dear SeqFan Members,

1- At https://en.wikipedia.org/wiki/Prime_number_theorem "In number theory<https://en.wikipedia.org/wiki/Number_theory>, the prime number theorem (PNT) describes the asymptotic<https://en.wikipedia.org/wiki/Asymptotic_analysis> distribution of the prime numbers<https://en.wikipedia.org/wiki/Prime_number> among the positive integers.";
2- Positive integers is a linear recurring sequence and the set of it’s elements form a group;
Now, considering a parabola  x=ay^2+by+C where a, b, and C are integers and a>0, (so, if there are negative elements they will be finite)
3- All sequences of the form x=ay^2+by+C are also linear recurring sequences and also form a group;
4- There is an isomorphism between each group aL^2+bL+C and integers group where coefficients (a;b;C) are integers ;
5- So, distribution of the prime numbers<https://en.wikipedia.org/wiki/Prime_number> among the positive integers or any sequence of integers of the form aL^2+bL+C are both asymptotic.
6- If it is true for a Parabola, will be true for any Polynomial.

Best regards,

Charles Kusniec
cel.: +55 11 987474974
<mailto:ck at centertap.com.br>ck at centertap.com.br<mailto:ck at centertap.com.br>

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