Primitive in Arithmetic Progression

Let S = {\(\dfrac{1}{r}:\)r = 1,2,3,...}. For all integer k>1, prove that there is a k-term arithmetic progression in S such that no addition term in S can be added to it to form a (k+1)-term arithmetic progression.

Prove that for every positive integers s,a,b with gcd(a,b) = 1, there are infinitely many integers n such that an + b  is product of s pairwise distinct prime numbers.

Let \(p_k\) be the k-th prime number. For every integer N, prove that there exists a positive integer k such that both \(p_{k-1}\) and \(p_{\left(k+1\right)}\) are not in the interval [\(p_k-N\),\(p_k+N\)].

Prove that if f(x) is pily nomial with rational coefficcients such that f(p) is a prime number for every prime number p, then either f(x) = x for all x or f(x) is the same prime constant for all x.

Prove that for every pair of positive integers n and N, there are consecutive positive intefers k,k+1,...,k+N such that \(\varphi\left(k\right),\varphi\left(k+1\right),...,\varphi\left(k+N\right)\)are all divisible by n, where \(\varphi\left(n\right)\) is as defined in Euler's theorem.

Des thte exist an inftinite squence of positive integers \(a_1,a_2,a_3,...\) such that \(a_m\) and \(a_n\) are copime if and only if |m-n| = 1?