4 problems

Consecutive-prime conjecture. For every fixed $\varepsilon>0$,

Let $c_0>1$. For fixed $c$ with $1<c<c_0$ and every sufficiently large positive integer $N$, consider the equation … where $p_1,p_2,p_3$ are prime numbers and each $x_i,y_i$ is an…

Let $k_0$ be a positive integer and let $c>1$ be non-integral. The Piatetski-Shapiro sequence is given by the integer parts $[n^c]$ for positive integers $n$. Consecutive Piatetski…

Let $f$ be a cusp form for the full modular group, let $1<c<8/7$, and let $\mathbbm{P}$ denote the set of primes. Write $\lambda_f(n)$ for the normalized Fourier coefficients of…