An entire function with two periods Can anybody help me with this question: 

If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?


If the function is entire, and has periods $2$ and $2i$, then it is bounded, and by Liouville, it is then a constant, so that any non-zero complex number is a period.
The above is well-known result in the early stages of the theory of elliptic functions. See, for example, Apostol's book "Modular Functions and Dirichlet Series in Number Theory" on page 5.
