4 problems

Let $p$ be an odd prime, let $m$ be a positive integer, and let $e$ be the relevant order parameter in the generalized cyclotomic construction. Let $\mathbf{s}^{\infty}$ be the gen…

The linear complexity conjecture. If $p$ is a non-Wieferich odd prime and $(s_i)$ is the sequence above, then its linear complexity is given by the formula stated in the source.

Linear-complexity conjecture. If $2^{p-1}\not\equiv 1\pmod {p^2}$, then the linear complexity $L$ of $s^\infty$ is

Let $p_{1},p_{2},n$ satisfy $\operatorname{gcd}(p_{1}-1,p_{2}-1)=4$, and suppose that $\frac{(p_{1}-1)(p_{2}-1)}{16}$ is odd. Let $p=2$ and let $q$ be a power of $2$. For common pr…