7 problems

For the Fibonacci recurrence and its parity transform modulo prime powers, Self-Similarity at Prime Powers. At each transition $p^k\to p^{k+1}$, all existing periods are preserved,…

Let $g(k)$ be the least positive integer such that for every pair of integers satisfying $g(k)<a_1<a_2$, there is a prime power $p^k\le g_{a_1,a_2}$ of the form $p^k=a_1x+a_2y$ wit…

Let $a_1$ and $a_2$ be integers with $a_2>a_1>40$ and $\gcd(a_1,a_2)=1$. Let $\pi_{2,a_1,a_2}$ denote the number of prime squares $p^2\le g_{a_1,a_2}$ representable as…

Let $n$ be a natural number greater than $23$. A prime power is a number $P^k$ where $P$ is prime and $k\geq 2$ is an integer. Main conjecture on limited prime power sums. Every na…

Let $k\geq 1$ be odd, and let $N$ be a sufficiently large even number. Prime-power plus almost-prime conjecture. Then $N$ can be expressed as the sum of a $k$th power of a prime an…

A pair of prime powers $(q_1,q_2)$ is a Hasse pair when … If both prime powers are odd, it is an odd Hasse pair. Hasse-pair coverage conjecture. Every prime power is a member of so…

Let $b$ and $c$ be non-zero integers such that $3\nmid bc$. For a sequence … let $D_{\operatorname{qr}}(n)$ denote its discriminator, the least modulus for which the first $n$ sequ…