We prove that if λ 1, λ 2, λ 3 and λ 4 are non-zero real numbers, not all of the same sign, λ 1/λ 2 is irrational, and π{variant} is any real number then, for any ε>0, the inequality |λ1p1+λ2p22+λ3p32+λ4p42+π{variant}|≤(maxjpj)-1/18+ε has infinitely many solutions in prime variables p 1,..., p 4. © 2012 Elsevier Inc.
A Diophantine problem with a prime and three squares of primes
LANGUASCO, ALESSANDRO;
2012
Abstract
We prove that if λ 1, λ 2, λ 3 and λ 4 are non-zero real numbers, not all of the same sign, λ 1/λ 2 is irrational, and π{variant} is any real number then, for any ε>0, the inequality |λ1p1+λ2p22+λ3p32+λ4p42+π{variant}|≤(maxjpj)-1/18+ε has infinitely many solutions in prime variables p 1,..., p 4. © 2012 Elsevier Inc.
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