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Take $z_0,z_1$ $\in \mathbb{C}$ to be $\mathbb{R}$ linearly independent. The exercise is to prove that if $f:\mathbb{C} \rightarrow \mathbb{C}$ is entire such that $f(z_0 +z) = f(z)$ and $f(z_1 +z) = f(z)$ $\forall z \in \mathbb{C}$ then $f$ is a constant function.

From the hypothesis I've managed to prove that $f(n.z_0 + m.z_1) = f(0)$ $\forall n,m \in \mathbb{Z}$. I feel that this should be enough to at least obtain a similar result for rational coefficients, however I am stuck. The best I was able to do was that if we take $ \gamma =\frac{p}{q}.z_0 +\frac{j}{k}.z_1$ with $p,q,j,k \in \mathbb{Z}$ then $f(\gamma.q.k) = f(0)$

Lucas G
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    Such functions are sometimes called "doubly-periodic". See https://math.stackexchange.com/questions/1244416/proving-that-a-doubly-periodic-entire-function-f-is-constant – Theo C. Apr 15 '24 at 01:23
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    from yesterday: https://math.stackexchange.com/questions/4898449/f-mathbbc-to-mathbbc-analytic-with-fz-fza-fzb – user8675309 Apr 15 '24 at 04:01

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