167 problems

Let $(a,b,c)\in\mathbb{N}^3$ be a triple of coprime integers satisfying $a+b=c$, and let $\operatorname{rad}(n)=\prod_{p\mid n}p$ denote the product of the distinct prime factors o…

Let $a,b,c$ be nonzero coprime integers satisfying $a+b+c=0$, let $N:=\operatorname{rad}(abc)$, and let $\omega(ab)$ denote the number of distinct prime factors of $ab$. Baker's ep…

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…

For a positive integer $m$, the double factorial $m!!$ is defined by … and … A solution of $a_1!!\cdots a_t!!=n!!$ is called nontrivial when it is not one of the solutions arising…

For an integer $m$, define its squarefree kernel, or radical, by … Let $a$, $b$, and $c$ be integers satisfying $a+b=c$ and $\operatorname{gcd}(a,b)=1$. The abc-conjecture. For eve…

Let $K$ be a number field or a one-dimensional function field of characteristic zero, and let $n\ge 3$. Let $[Z_1:\cdots:Z_n]$ be the standard homogeneous coordinates on…

Erdős's conjecture. The equation has only finitely many integer solutions.

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

The abc conjecture. As a consequence of the abc conjecture, this equation has only finitely many such solutions, except for $1^r+2^3=3^2$.

For positive integers $a,b,c$ with $a+b=c$ and $a,b,c$ pairwise coprime, let $r(abc)$ denote the product of the distinct primes dividing $abc$. The abc-conjecture. One has … The so…

For an integer $n$, let $\gamma(n)$ denote its largest squarefree divisor. The abc conjecture. For every $\epsilon>0$, there are only finitely many integers $a$, $b$, and $c$ satis…

Let $K$ be a number field. For a nonzero triple $(a,b,c)$ of algebraic numbers in $K$ satisfying $a+b=c$, write $h_K(a:b:c)$ for the logarithmic Weil height and…

Let $K$ be a number field. For $a,b,c\in K\setminus\{0\}$ with $a+b=c$, let $h_K(a:b:c)$ denote the absolute logarithmic height of $(a:b:c)$, and let $\operatorname{rad}(a:b:c)$ de…

ABC conjecture. For every $cepsilon>0$,

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 positive integers $a$ and $b$, let $B$ denote the dyadic scale of the larger variable, so that $\max(a,b)\in[B,2B)$, and define … where $\gamma(n)=\prod_{p\mid n}p$ and…

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…

The abc conjecture. For every $\epsilon>0$, there exists a constant $C(\epsilon)$ such that for all $(a,b,c)\in\mathbb{N}^3$ with $\gcd(a,b,c)=1$ and $a+b=c$,

Weak effective $abc$ conjecture. There are no relatively prime triples of positive integers $(a,b,c)$ satisfying $a+b=c$ and

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…

Let an $abc$-triple be a triple of positive coprime integers $(a,b,c)$ satisfying $a+b=c$, and let $q_s(a,b,c)$ denote its standard quality. The $abc$-conjecture. For every real…

For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$ such that for all coprime nonzero integers $a$, $b$, and $c$ satisfying $a+b=c$ one has … where…

ABC conjecture. For each $\varepsilon>0$, there is a constant $K_{\varepsilon}$ such that whenever $a,b,c$ are nonzero integers with $\gcd(a,b,c)=1$ and $a+b=c$,