# Sequences containing infinitely many primes

What are some interesting sequences that contain infinitely many primes?

If it takes form of a polynomial, Dirichlet's theorem answer the question completely for linear polynomial. What about polynomials of degree more than 1? Is there a known polynomial of degree more than 1 that contains infinitely many primes?

What about more complicated sequences like $2^n+3^n$, $n!+1$, etc?

Please provide examples that are as interesting as possible, accompanied with proofs (or reference to proofs) if not too difficult.

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Bunyakovsky conjecture –  pedja Feb 6 '12 at 10:17

A polynomial formula for the primes (with 26 variables) was presented by Jones, J., Sato, D., Wada, H. and Wiens, D. (1976). Diophantine representation of the set of prime numbers. American Mathematical Monthly, 83, 449-464.

The set of prime numbers is identical with the set of positive values taken on by the polynomial

$(k+2)(1-(wz+h+j-q)^2-((gk+2g+k+1)\cdot(h+j)+h-z)^2-(2n+p+q+z-e)^2-(16(k+1)^3\cdot(k+2)\cdot(n+1)^2+1-f^2)^2-(e^3\cdot(e+2)(a+1)^2+1-o^2)^2-((a^2-1)y^2+1-x^2)^2-(16r^2y^4(a^2-1)+1-u^2)^2-(((a+u^2(u^2-a))^2-1)\cdot(n+4dy)^2+1-(x+cu)^2)^2-(n+l+v-y)^2-((a^2-1)l^2+1-m^2)^2-(ai+k+1-l-i)^2-(p+l(a-n-1)+b(2an+2a-n^2-2n-2)-m)^2-(q+y(a-p-1)+s(2ap+2a-p^2-2p-2)-x)^2-(z+pl(a-p)+t(2ap-p^2-1)-pm)^2)$

as the variables range over the nonnegative integers.

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Haloo

Consider system of intervals: [A1,+∞); [A2,+∞); [A3,+∞); ... A1

Let NP(i) denote the number of primes of the form "s^2+d" , (+infinity is also acceptable) , "d" is constant integer, occurring in interval [Ai,+∞), i=1,2,3, ... I have proved NP(1)+NP(2)+NP(3)+...= +∞.

Is deduction that there has to exist such number k that NP(k)=+∞ OK ?.

Note that intersection of all closed intervals [Ai,+∞), i=1,2,3, ... is null set.

I do not have satisfactory proof of upper statement. Otherwise I would prove , that exist infinitely many constants "d" such that sequence s^2+d contains infinitely many primes. "x^2" could by substitutes by any polynomial.

I would like to publish this result and I looking for somebody who is willing to collaborate and who is native English. Thanks for Your answer Sincerely Pavol Galik pavol.galik01@gmail.com

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