Author

R. Laniewski

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Preprint Sep 2026

Weak Hall's conjecture, approximation gains, and continued fractions

For an integer solution of $y^2=x^3+k$ with $x,y\ge 1$ and $k\ne 0$, the integers $x^3$, $|k|$ and $y^2$ form a triple in which one term is the sum of the other two. When $\gcd(x,y)=1$ this triple is coprime, and the $abc$ conjecture predicts that its quality is asymptotically at most $1$. Following M\"uller, Taktikos...

R. Laniewski, K. Muller · 0 citations

Preprint Aug 2026

Radical defects, Wieferich primes, and the $abc$ conjecture

For coprime $a+b=c$, the parity class in which $a$ and $b$ are odd and $c$ is even admits an exact treatment. Triples are measured by the radical excess $E_{\varepsilon}=\log c-(1+\varepsilon)\log\operatorname{rad}(abc)$, and called transgressive at fixed $\varepsilon>0$ when it is non-negative. The $abc$ conjecture as...

R. Laniewski · 0 citations