Can you prove or disprove the following claim:

>Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime.

You can run this test [here][1]. I have verified this claim for all composite $N$ up to $2^{100} \cdot 3^{100}+1$ with $2 \le c \le 100$ , and for all prime $N$ from [this][2] list.


  [1]: https://sagecell.sagemath.org/?z=eJzzszW35krLL9JItDXSyc_XSbLNyUwr0fDNT9FI1PHTjNPQ8NM11NQ309S0zkwDCxeX5mpk2hroGOnAVeomgZRmamoCKVugVEFRZl6JhhKQyk1V0rROKkpNzAYaAACX4B9f&lang=gp&interacts=eJyLjgUAARUAuQ==
  [2]: https://oeis.org/A058383