Let $ k \geqslant 2 $ be an integer. We studied the number of integers which form perfect $ k $-th powers in the way of
$ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k. $
For $ k \geqslant4 $, we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for $ k = 2 $.
Citation: Tingting Wen. On the number of integers which form perfect powers in the way of $ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k $[J]. AIMS Mathematics, 2024, 9(4): 8732-8748. doi: 10.3934/math.2024423
Let $ k \geqslant 2 $ be an integer. We studied the number of integers which form perfect $ k $-th powers in the way of
$ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k. $
For $ k \geqslant4 $, we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for $ k = 2 $.
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Tingting Wen. On the number of integers which form perfect powers in the way of $ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k $[J]. AIMS Mathematics, 2024, 9(4): 8732-8748. doi: 10.3934/math.2024423
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