Can you prove or disprove the following claim:

>Let $n$ be a natural number of the form $4k+3$ , and let $c$ be the smallest odd prime such that $\binom{c}{n}=-1$ , where $\binom{}{}$ denotes a Jacobi symbol. Let $z=c+i$ be a complex number , then $n$ is prime iff $\text{Re}\left(z^n\right) \equiv c \pmod n$  and $\text{Im}\left(z^n\right) \equiv -1 \pmod n $ .

You can run this test [here][1]. I have verified this claim for all $n$ up to $5 \cdot 10^{11}$.


  [1]: https://sagecell.sagemath.org/?z=eJxNjdEKwjAMRd_7GfMl0fZhKvhQ-gn-wkC6FspcMrMKA_HfbTeEQQjcc5N7D4rc9WYV9_0kaQxA6D6RZRPeXTSzThEGYQp-CAJelxNnWi0hv4XAI6L9qvkl2cOjmnfuYdHr7s6mIrSqZDxTzFDxruy0_ZU4wjJdzd7bQKbF46ILoAzNyhv8S8_jxHPKBaH9AfcKQTg=&lang=gp&interacts=eJyLjgUAARUAuQ==