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Pairs of additive equations. IV: Sextic equations. (English) Zbl 0532.10008

In this paper the author continues his study of the simultaneous solubility of two homogeneous additive equations with integer coefficients [Mich. Math. J. 19, 325-331 (1972; Zbl 0244.10046); J. Number Theory 17, 80-92 (1983; Zbl 0516.10043), Proc. Edinb. Math. Soc. 26, 191-211 (1983; Zbl 0497.10036)]. It is shown that the sextic equations \[ a_ 1 x_ 1^ 6+...+a_ N x_ N^ 6=0,\quad b_ 1 x_ 1^ 6+...+b_ N x_ N^ 6=0 \] have a non-trivial simultaneous solution in integers, providing that \(N\geq 73=2\cdot 6^ 2+1\), that there are non-singular real, 2-adic and 3-adic solutions and that the N ratios \(a_ i/b_ i\) are reasonably distributed. This verifies for the case \(k=6\) a remark of H. Davenport and D. J. Lewis [Proc. Sympos. Pure Math. 12, 74-98 (1969; Zbl 0226.10026)] that a form of Artin’s conjecture should hold for pairs of homogeneous additive equations of degree k; results of this type have already been established for \(k\leq 5\) and \(k\geq 18\).
Reviewer: M.M.Dodson

MSC:

11D72 Diophantine equations in many variables
11P55 Applications of the Hardy-Littlewood method
11P05 Waring’s problem and variants
11D41 Higher degree equations; Fermat’s equation
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