Open problems in number theory

Open problems in number theory (continued):

I try to list some open problems in number theory:

(1)

Goldbach’s conjecture: Any even number can be written as a sum of two primes.
28 = 5 + 23 = 11 + 17
168 = 5 + 163

(2)
Twin primes: There are infinitely many primes p such that p + 2 is also prime.
Examples : (3, 5) (5, 7) (11, 13) (17, 19) (29, 31) (41, 43)

(3)Landau’s conjecture: There are infinitely many primes of the form n^2 + 1.

Schinzel’s hypothesis: Let f(X) and g(X) be two irreducible polynomials in Z[X]. Suppose there is no integer n such that n divides f(k) · g(k) for all k. Then there are infinitely many values of k such that f(k) and g(k) are both prime numbers.

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Theorem of Terence Tao and Ben Green: There are arbitary long arithmetic progression in the primes.