I am having trouble finding a number where 579^$65$ is congruent to x mod 679 and x has to be less than 676. i did the trick of 2's and got:
$579^2$ $\equiv$ 494 mod 679
$494^2$ $\equiv$ 275 mod 679
$275^2$ $\equiv$ 256 mod 679
$256^2$ $\equiv$ 352 mod 679
$352^2$ $\equiv$ 326 mod 679
$326^2$ $\equiv$ 352 mod 679
i notice the pattern at the end but what does that exactly mean?
What's happening is that $\mathbb{Z}/679\mathbb{Z}^{\times} \cong \mathbb{Z}/7\mathbb{Z}^{\times}\times \mathbb{Z}/97\mathbb{Z}^{\times} \cong \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/96\mathbb{Z}$ contains several elements of order $3$ (note $3 | 6$ and $3 | 96$), i.e. elements $x$ with $x^3 = 1$ mod $679$, which implies $(x^2)^2 = x^4 = x$.
You've found two of those: $352$ and $326$.
