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

israel at math.ubc.ca israel at math.ubc.ca
Tue May 16 17:44:54 CEST 2017

```Nonsense. For example, there are no primes in the sequence n^2 + 2 n + 1.
Bunyakovsky's conjecture states that a polynomial f(n) has infinitely many
primes if 1) it is irreducible, 2) its leading coefficient is positive, and
3) f(n) have no common prime factor. But this is still very much open. In
fact, we don't even know a single nonlinear polynomial that provably
contains infinitely many primes.

Cheers,
Robert Israel

On May 16 2017, Charles Kusniec wrote:

>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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