Let $(x_{k})_{k≥2}$ and $(y_{k})_{k≥2}$ be two non constant sequences of strictly increasing positive integers such that $x_{k}>1,y_{k}>1$ for all $k≥2$. I want to get a counterexample to the following claim:
Claim: If there exist one positive integer $m≥2$ such that there is a prime number in the interval $(y_{m},x_{m}+y_{m})$, then for all $k≥m$ the interval $(y_{k},x_{k}+y_{k})$ contain a prime number.
I have remark that if $x_{k}=(k+1)²- k²,y_{k}=k²$ then the problem is similar to the Legendre conjecture.
$$x_2 = 2, y_2 = 22, x_3 = 3,y_3 = 25$$ $$x_k = x_{k-1} +1 \ and \ y_k = y_{k-1} +3 \ \forall \ k \ge 3$$ For k = 2 we have a prime 23 but for k = 3 there no prime in the range (25,28)