# A detail oriented prime conjecture

Conjecture For arbitrary integers $$\ 0 \le k \le m\$$ there exists integer $$\ n\ge m\$$ such that for every natural number $$\ s\$$ at least one of the numbers $$\ p(x)+s\ (\text{where}\ k\le x\le n)\$$ is not prime.

Here, $$\ p(0)=2, p(1)=3,\ldots\$$ is the strictly increasing sequence of all prime numbers.

Let $$q=p(k)$$. Using Dirichlet's theorem on primes in arithmetic progressions, there is $$n\geq m$$ large enough so that for any $$a$$ not divisible by $$q$$, there is some $$k\leq x\leq n$$ such that $$p(x)\equiv a\pmod q$$. Also taking $$a=0$$ and $$x=k$$, we see this is true for all residue classes mod $$q$$.
Take any natural $$s$$ (which presumably also requires $$s>0$$ for you). Pick some $$k\leq x\leq n$$ such that $$p(x)\equiv -s\pmod q$$ (which exists by construction). Then $$p(x)+s$$ is divisible by $$q$$ and greater than $$q$$.
• @WlodAA Do you mean including $k=0$? I don't see how that makes any difference really. My solution should still work. – Wojowu Sep 20 at 9:04