An Additive Diophantine Inequality with Mixed Powers
Let k ∈ {5,6} and η be any given real number.Suppose that λ1,λ2,...,λ7 are nonzero real numbers,not all of the same sign and λ1/λ2 is irrational.It is proved that the inequality |λ1x21+λ2x32+λ3x33+λ4x34+λ5x35+λ6x46+λ7xk7+η|＜(max1≤j≤7Xj)-σ has infinitely many solutions in positive integers x1,x2,...,x7 for 0＜σ＜1/12(k-3)·This result constitutes an improvement upon that of Li and Gong.
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