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- W4291006679 abstract "Let $$p>5$$ be a fixed prime and assume that $$alpha _1,alpha _2,alpha _3$$ are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form $$alpha _1x_1^2+alpha _2x_2^2+alpha _3x_3^2equiv 0bmod {q}$$ with $$q=p^n$$ , where $$max {|x_1|,|x_2|,|x_3|}le N$$ and $$(x_1x_2x_3,p)=1$$ . (In fact, we consider a smoothed version of this problem.) If $$alpha _1,alpha _2,alpha _3$$ are fixed and $$nrightarrow infty $$ , we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $$Ngg q^{1/2+varepsilon }$$ . If these coefficients are allowed to vary with n, we show that this formula holds if $$Ngg q^{11/18+varepsilon }$$ . The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $$N gg q^{5/8+varepsilon }$$ for odd square-free moduli q." @default.
- W4291006679 created "2022-08-13" @default.
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- W4291006679 date "2022-08-11" @default.
- W4291006679 modified "2023-09-27" @default.
- W4291006679 title "Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers" @default.
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- W4291006679 doi "https://doi.org/10.1007/s40993-022-00360-7" @default.
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