12 problems

Infinitude conjecture. $E^*_{Y^2}$ contains infinitely many numbers of the form $\pi_1\pi_2$ for such primes $\pi_1$ and $\pi_2$.

Fix a prime $q\equiv1\,(\mathrm{mod}\,4)$, an integer $r\geq2$, and $\mathbf{a}=(a_1,\dots,a_r)\in\mathbb{N}^r$. Let $E_n$ be the increasing sequence of sums of two squares, and de…

For $k\geq1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ and…

For $k\geq 1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ be the nor…

Let $\mathcal{H}^*=\{h_1,h_2,\ldots\}$ be an ordered set of integers. It is admissible if every finite truncation $\{h_1,\ldots,h_k\}$ is admissible, meaning that its reduction mod…

Let $b(n)$ be the indicator of integers representable as sums of two squares. Let $B(x)=\sum_{n\leq x}b(n)$, let $I(x;H)$ be the paper's smooth approximation to $B(x+H)-B(x)$, and…

Gap conjecture for sums of two squares. For every $\varepsilon>0$,

Let $Q(x,y):=Ax^2+2Bxy+Cy^2+Dx+Ey+F$ be a binary quadratic form with integral coefficients satisfying … and with discriminant … for a prime $q=4k+1$. Suppose that $Q(x,y)$ is posit…

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…

Let $\mathbb{S}$ denote the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…

Let $a$ be a positive integer that is not a perfect power, meaning $a\ne c^k$ for every positive integer $c$ and every $k>1$. Let $m$ be the smallest positive integer such that…