50 problems

Congruence ABC conjecture for $N$. For each $\epsilon>0$, there exists a constant $C_\epsilon$ such that

Let $K$ be a number field, let $M_K$ be its set of places, and define the height $\operatorname{h}$ and conductor $\operatorname{cond}$ as in the preceding number-field abc formula…

The abc-conjecture for a number field. For every $\kappa>1$ and every $x\in K$ distinct from $0$ and $1$,

For $n\in\mathbb{N}$, let $\gamma(n)=\prod_{p\mid n}p$ and let $\omega(n)$ denote the number of distinct prime factors of $n$. Define … The convention $0^0=1$ is used when $n=1$. T…

For coprime positive integers $a$ and $b$, put $c=a+b$ and define … where $\gamma(n)=\prod_{p\mid n}p$ is the radical of $n$. Conjecture A. There exists a real number $C_1$ such th…

For an $abc$-triple $(a,b,c)$, let $q_C(a,b,c;\alpha,\beta)$ be the alternative quality metric considered in the source, with parameters $\alpha$ and $\beta$. Generalized quality m…

Classical abc conjecture. For every $\varepsilon>0$, there is a constant $K(\varepsilon)$ depending only on $\varepsilon$ such that whenever $a$, $b$, and $c$ are three coprime non…

ABC conjecture. For every $\epsilon>0$, there is a constant $C_\epsilon$ such that

The abc conjecture for a Galois number field. For every $\epsilon>0$,

For a non-zero integer $m$, let $\operatorname{rad}(m)$ be its largest square-free positive divisor. For $n\geq 3$ and $a=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ with all $a_i\neq 0$, def…

Let $S$ be a finite set of primes. Consider integer solutions $(a,b,c,x,y,z,p,q,r)$ of the generalized Fermat equation $eqref{eq:genfermat}$ satisfying $\gcd(ax,by,cz)=1$, $p,q,r>0$…

Let $a,b,c$ be nonzero integers, and let $(x,y,z,p,q,r)$ be integers satisfying the generalized Fermat equation $eqref{eq:genfermat}$, with $\gcd(x,y,z)=1$, $p,q,r>0$, and…

Let $n\geq 3$. For an $n$-tuple $a=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ with nonzero entries, write … and for a sequence $A=\{a^{(1)},a^{(2)},\ldots\}$ of such tuples write…

Let $n\ge3$. For non-zero integers $x_1,\ldots,x_n$, write $\operatorname{rad}(x_1x_2\cdots x_n)$ for the largest positive squarefree divisor of their product. The strong n-terms a…

Let $n\ge3$. For integers $x_1,\ldots,x_n$, write $\operatorname{rad}(x_1x_2\cdots x_n)$ for the largest positive squarefree divisor of their product. The n-terms abc conjecture. T…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and discriminant $\Delta_K$. For $\epsilon>0$ and a non-zero integral ideal $\mathfrak{a}$ of $\mathcal{O}_K$, let…

Let $A$, $B$, and $C$ be positive integers such that $A+B=C$ and $(A,B)=1$. Define $\operatorname{rad}(ABC)$ to be the product of the distinct prime divisors of $ABC$. Explicit ABC…

Let $K$ be a number field and let $I$ be an ideal of $\mathcal{O}_K$. Define … where $N(\mathfrak{p})=\#(\mathcal{O}_K/\mathfrak{p}\mathcal{O}_K)$. Generalized abc conjecture. For…

The abc conjecture. For each positive real number $\varepsilon$ there is a positive number $c(\varepsilon)$, which depends on $\varepsilon$ only, such that for all pairwise coprime…

Let $q=a/b$ be a nonzero rational number, let $\varphi_q:\overline{\operatorname{Spec} \mathbb Z}^{\mathrm{Smi}}\to\overline{\operatorname{Spec} \mathbb Z}^{\mathrm{Smi}}$ be the a…

Infinitude conjecture for $\mu$-hits. There are infinitely many $\mu$-hits.

Let $E$) be an elliptic curve over $\mathbb{Q}$, with conductor $N=N(E)$ and Tamagawa product $\tau=\tau(E)$. Here $\ll$ denotes an absolute implied constant, and $\epsilon>0$ is a…

The abc-conjecture consequence. If $D\neq 0$, then one has the lower bound

Monomial-difference lower-bound conjecture. If $D e 0$, then

A positive integer is powerful if every prime divisor $p$ satisfies $p^2\mid n$. For an integer $N$, the three consecutive integers are $N$, $N+1$, and $N+2$. Erdős's conjecture. T…