6 problems

Let $f(x)$ and $g(x)$ be irreducible polynomials of degree $k$ and exponent $e$, and suppose that $2k=nm$, where $n>1$ and $m>1$, and $e=rt$ with $\operatorname{gcd}(r,t)=1$. Foldi…

Let $p$ be an odd prime, and let $\mathcal L=(\ell_n)_{n\geq 0}$ be the $p$-periodic Legendre sequence associated with $f(X)=X$, with $\Phi_2(\mathcal L,N)$ its $N$th $2$-adic comp…

Let $\mathcal S=(s_n)_{n\geq 0}$ be one of the Thue–Morse sequences, the Rudin–Shapiro sequences along the values of $f(X)\in\{X,X^2,X^3,X^4\}$, or the binary sequence of the Zecke…

Let $a(n)$ be an automatic sequence, let $\alpha$ be a symbol in its alphabet, and let $c\in(1,2)$. Write $d_{\log}(a(n),\alpha)$ and $d(a(n),\alpha)$ for the logarithmic and ordin…

Let $(a_n)_{n\geq 1}$ be the binary Ehrenfeucht–Mycielski sequence, and let $N_0(n)$ and $N_1(n)$ denote the numbers of $0$s and $1$s, respectively, among its first $n$ bits. Balan…

A Paley–Hadamard difference set has length $n$. Golomb's conjecture. If … then $n$ should either be a prime number, a product of twin primes, or have the form … where $k$ is a posi…