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In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]