The integral part of a nonlinear form with a square, a cube and a biquadrate

Abstract

In this paper, we show that if λ1,λ2,λ3{\lambda }_{1},{\lambda }_{2},{\lambda }_{3} are non-zero real numbers, and at least one of the numbers λ1,λ2,λ3{\lambda }_{1},{\lambda }_{2},{\lambda }_{3} is irrational, then the integer parts of λ1n12+λ2n23+λ3n34{\lambda }_{1}{n}_{1}^{2}+{\lambda }_{2}{n}_{2}^{3}+{\lambda }_{3}{n}_{3}^{4} are prime infinitely often for integers n1,n2,n3{n}_{1},{n}_{2},{n}_{3}. This gives an improvement of an earlier result.

Keywords

• diophantine inequalities
• primes
• davenport-heilbronn method
• 11p32
• 11p55
• 11j25