# 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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Look here

http://mathworld.wolfram.com/MillsConstant.html

The formula is, unfortunately, of little practical value because the constant must be known to tremendous many digits to produce $15-20$ primes, and the sequence grwos too fast, but theoretically, it is beautiful.

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