Prove that at least one of those numbers is a prime

Question

Conjecture:

Given a natural number $n>1$. Then there is a prime in the sequence $$n+1,2n+1,\dots,(n-1)n+1$$

Tested for all $n<10,000,000$.

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I found this: http://oeis.org/A034693

Answer 1

Dirichlet's theorem on arithmetic progressions states that, if $\gcd(a,n)=1$ then there exist infinitely many primes in the arithmetic progression $a,a+n,a+2n,a+ 3n,\dots$

Linnik proved that there are constants $c_1$ and $c_2$ such that the least prime $p(a,n)$ which is congruent to $a$ modulo $n$ satisfies $p(a,n)\leq c_1 n^{c_2}$. Details about these constant $c_1$ and $c_2$ can be found here: https://mathoverflow.net/questions/80865/least-prime-in-a-arithmetic-progression

Your conjecture says that $p(1,n)\leq (n−1)n+1$ for any integer $n>1$.