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On a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality | λ p + λ p ² + λ p k + ϖ | ( m a x j p j ) - ( 33 - 29 k ) / ( 72 k ) + ε has infinitely many solutions in prime variables p₁, p₂, p₃.

On an additive problem of unlike powers in short intervals

Qingqing Zhang (2022)

Czechoslovak Mathematical Journal

We prove that almost all positive even integers n can be represented as p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 4 N | N 1 - 1 / 54 + ε for 2 k 5 . As a consequence, we show that each sufficiently large odd integer N can be written as p 1 + p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 5 N | N 1 - 1 / 54 + ε for 1 k 5 .

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

Let q [ t ] be the polynomial ring over the finite field q , and let N be the subset of q [ t ] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A N for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D ( N ) q N ( l o g N ) 7 / N .

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