Diophantine approximation with one prime, two squares of primes and one kth power of a prime
Let 1<k<14/51\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5, λ1,λ2,λ3{\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ4{\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ1/λ2{\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and...
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
De Gruyter
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/d0891e1ae9dc406db57be3be877d3cbc |
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| Sumario: | Let 1<k<14/51\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5, λ1,λ2,λ3{\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ4{\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ1/λ2{\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and let ω\omega be a real number. We prove that the inequality ∣λ1p1+λ2p22+λ3p32+λ4p4k−ω∣≤(max(p1,p22,p32,p4k))−ψ(k)+ε| {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables p1,p2,p3,p4{p}_{1},{p}_{2},{p}_{3},{p}_{4} for any ε>0\varepsilon \gt 0, where ψ(k)=min114,14−5k28k\psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right). |
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