We prove that the uniform $abc$-conjecture for number fields implies that there are no Siegel zeros for $L$-function of quadratic characters $(-d/.)$, by studying the singular moduli that give rise to solutions of $j(\tau)=\gamma_2(\tau)^3=\gamma_3(\tau)^2+1728$.

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We show that for every fundamental discriminant $D>3705$ there is a prime $p<\sqrt{D}/2$ for which $(D/p)=-1$.

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We show that there is a constant $c\in (\frac 12,1)$ such that $\sim cp$ zeros of the $p$th Fekete polynomial lie on the unit circle

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We bound from below the lcm of $k$ integers from a short interval. This is then used to bound the length of any arc of the hyperbola $xy=N$ which contains $k$ lattice points.

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