13 problems

Borwein–Choi conjecture.

Let a ${\mathcal B}$-number be a positive integer represented by the binary quadratic form $a^2+ab+b^2$. Suppose that such a number is written in the general form … where $x$ has n…

Let $C,D$ be distinct, nonzero rational numbers, and let $f(x,y)=Cx^2+Dy^2$ be a binary quadratic form. A rational periodic vector of type $(L,R,L,R)$ is a rational vector undergoi…

Let $m$ be a natural number, and consider the minimal triples associated with the generalized Markoff equation studied in the paper. Minimal-triple growth conjecture. For a given…

Let $k$ be a positive integer such that $4\Vert k$ and $2k-1$ is a prime power. Write … where $a$ is square-free. The form $(a^2b^2,0,2k-1)$ is the binary quadratic form…

Miller's class-number conjecture. The sum of these class numbers over primes should satisfy

Hirschhorn's conjecture. Hirschhorn conjectured that Ramanujan's “amazing" identity in his “Lost Notebook" could be proved via this Type$(4)$ identity. He later derived the identit…

Bounded-error conjecture. There is a small constant $M$ such that, for all $k$ and $d$,

Let $p$ be an odd prime represented as … Here $x,y$ are integers, and $[t]$ denotes the integer part of $t$. The $2x^2+29y^2$ congruence. The paper conjectures that … This is one o…

The paper considers Theorems 6.1–6.4 and 7.1–7.2, whose hypotheses include the alternative coprimality condition $(c,x+d)=1$ or $(d_0,x+c)=1$. Removal conjecture. The condition … c…

Let the hypotheses of Theorem 5.2 and Corollary 5.2 include the alternative coprimality condition $(c,x+d)=1$ or $(d_0,x+c)=1$. Removal conjecture. The condition … can also be canc…

The paper considers the theorems and corollaries numbered 4.1–4.5, whose hypotheses include the alternative coprimality condition $(c,x+d)=1$ or $(d_0,x+c)=1$. Removal conjecture.…

Ramanujan's conjecture. The sequence $\lambda(1,7;n)$ is multiplicative, and, for $s>1$,