is it true? $\operatorname{round}(\varphi^{p_n}) \equiv 1 \!\!\pmod{p_n} $ where $p_n$ is $n$th prime number and $\varphi=1.618033...$ The $\operatorname{round}(\varphi^{n})$ is called n-th Lucas number.
These numbers have most interesting features. One of them is the relationship with prime numbers for example this article
In this regard, I happened to see that :
$$\operatorname{round}(\varphi^{n})\equiv 1\!\!\!\!\pmod{n}$$
if and only if $n$ is prime number.
I wnat to know, is it proven? or it has some exceptions in the larger numbers?

Let $L_n$ be the $n$-th Lucas number. We have $$\text{round}(\varphi^n) = L_n$$ for $n > 2$ (see here). We indeed have
$$L_n \equiv 1 \!\!\pmod n$$
if $n$ is prime (see here), but there are also composite numbers $n$ for which above congruence is true, and these are the Fibonacci Pseudoprimes.
