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### A Bound for Prime Solutions of Some Ternary Equations.

Mathematische Zeitschrift

Acta Arithmetica

### A Diophantine inequality with four squares and one $k$th power of primes

Czechoslovak Mathematical Journal

Let $k\ge 5$ be an odd integer and $\eta$ be any given real number. We prove that if ${\lambda }_{1}$, ${\lambda }_{2}$, ${\lambda }_{3}$, ${\lambda }_{4}$, $\mu$ are nonzero real numbers, not all of the same sign, and ${\lambda }_{1}/{\lambda }_{2}$ is irrational, then for any real number $\sigma$ with $0<\sigma <1/\left(8\vartheta \left(k\right)\right)$, the inequality $|{\lambda }_{1}{p}_{1}^{2}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{2}+\mu {p}_{5}^{k}+\eta |<{\left(\underset{1\le j\le 5}{max}{p}_{j}\right)}^{-\sigma }$ has infinitely many solutions in prime variables ${p}_{1},{p}_{2},\cdots ,{p}_{5}$, where $\vartheta \left(k\right)=3×{2}^{\left(k-5\right)/2}$ for $k=5,7,9$ and $\vartheta \left(k\right)=\left[\left({k}^{2}+2k+5\right)/8\right]$ for odd integer $k$ with $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).

Acta Arithmetica

### A new form of the circle method, and its application to quadratic forms.

Journal für die reine und angewandte Mathematik

Acta Arithmetica

### A Note on the Exceptional Set for Goldbach's Problem in Short Intervals.

Monatshefte für Mathematik

Acta Arithmetica

### A propos du problème de Waring

Seminaire de Théorie des Nombres de Bordeaux

Acta Arithmetica

### A short intervals result for linear equations in two prime variables.

Given A and B integers relatively prime, we prove that almost all integers n in an interval of the form [N, N+H], where N exp(1/3+e) ≤ H ≤ N can be written as a sum Ap1 + Bp2 = n, with p1 and p2 primes and e an arbitrary positive constant. This generalizes the results of Perelli et al. (1985) established in the classical case A=B=1 (Goldbach's problem).

### A sieve approach to the Waring-Goldbach problem. I. Sums of four cubes

Annales scientifiques de l'École Normale Supérieure

Acta Arithmetica

### A ternary Diophantine inequality over primes

Acta Arithmetica

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)].

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Additive representation in thin sequences, I : Waring's problem for cubes

Annales scientifiques de l'École Normale Supérieure

Acta Arithmetica