数学季刊 ›› 2015, Vol. 30 ›› Issue (4): 545-554.doi: 10.13371/j.cnki.chin.q.j.m.2015.04.007
摘要: It is proved that if λ1, λ2, ···, λ7are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 < σ <1/16, the inequality |λ1x12+ λ2x22+∑7 i=3λixi4+ η| <( max1≤i≤7|xi|)-σhas infinitely many solutions in positive integers λ1, λ2, ···, λ7 Similar result is proved for |λ1x12+ λ2x22+ λ3x32+ λ4x44+ λ5x54+ λ6x64+ η| <( max1≤i≤6|xi|)-σ.These results constitute an improvement upon those of Shi and Li.
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