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

Quanwu Mu, Minhui Zhu, Ping Li (2019)

Czechoslovak Mathematical Journal

Let k ≥ 5 be an odd integer and η be any given real number. We prove that if λ 1 , λ 2 , λ 3 , λ 4 , μ are nonzero real numbers, not all of the same sign, and λ 1 / λ 2 is irrational, then for any real number σ with 0 < σ < 1 / ( 8 ϑ ( k ) ) , the inequality | λ 1 p 1 2 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 2 + μ p 5 k + η | < max 1 ≤ j ≤ 5 p j - σ has infinitely many solutions in prime variables p 1 , p 2 , ⋯ , p 5 , where ϑ ( k ) = 3 × 2 ( k - 5 ) / 2 for k = 5 , 7 , 9 and ϑ ( k ) = [ ( k 2 + 2 k + 5 ) / 8 ] for odd integer k with k ≥ 11 . This improves a recent result in W. Ge, T. Wang (2018).

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

M. B. S. Laporta (1997)

Revista Matemática de la Universidad Complutense de Madrid

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 ternary Diophantine inequality over primes

Roger Baker, Andreas Weingartner (2014)

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 - η holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

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