For a given $n$, i am trying to find a constant value $c$ such that you can always find a prime $p$ of the form $4x+1$ for some $x \in \mathbb{Z}_+$ and $n < p < cn$. I want to find the smallest such $c$ for which the above property holds for all values of $n$.

I searched in literature, i could only find that $(n!)^2+1$ has a prime divisor $p$ of the form $4x+1$ and $n < p < (n!)^2+1$. But $n!$ upper bound is huge for my need.

Also i found that we can find a prime $p$ of the form $4x+1$ such that $n< p < (p_1...p_k)^2+1$ where $p_1,...,p_k$ are all the prime numbers of the form $4x+1$ and less than $n$. This upper bound of $(p_1...p_k)^2+1$ is also large for my need.

Also i found some asymptotic formulas but i want formulas which holds for all numbers.

It would be great if i can find a constant $c$ such that there is always a prime of the form $4x+1$ between $n$ and $cn$ and the property holds for all $n$.

Thanks in advance.