8 problems

Tunnell's conjecture. The integer $n$ is a congruent number if and only if

Hsia's conjecture. If $f$ and $g$ primitively represent the same integers, then they are equivalent.

Kaplansky's conjecture. If $Q(f)=Q(g)$, then one of the following holds: $f\simeq g$; $f$ and $g$ are both regular; for some $a,b\in\mathbb{Z}$,

Sun's conjecture. Every natural number $n$ can be written as

Farhi's conjecture. For each integer $m\geq 3$, every natural number $n$ can be represented as

Let $L$ and $\hat{L}$ be the two lattices in the prehomogeneous vector space whose zeta functions are $\xi_i(L,s)$ and $\xi_i(\hat{L},s)$ for $i=1,2,3$, as defined in the paper. Oh…

Let $n$ be a nonnegative integer, and let $x,y,z$ be integers. Universal $7n+1$ representation conjecture. Every number of the form $7n+1$ can be represented as … This conjecture i…

Let $\bar{B}(n)$ be the sequence defined in the paper, and let $X$ tend to infinity. Positive density conjecture for $\bar{B}(n)$. There exists a constant $\sigma_2\in(0,1]$ such t…