For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies$${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound$${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.

Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality $|p{₁}^c+p{₂}^c+p{₃}^c-R|<R^{-\eta }$ holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

We show how to adapt Terr’s variant of the baby-step giant-step algorithm of Shanks to the computation of the regulator and of generators of principal ideals in real-quadratic number fields. The worst case complexity of the resulting algorithm depends only on the square root of the regulator, and is smaller than that of all other previously specified unconditional deterministic algorithm for this task.