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By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.
We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let
.
We obtain the asymptotic formula
S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X),
where C₁,C₂ are two constants. This improves upon the error term obtained by Guo and Zhai (2012).
Let denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When or 4, it is established that the anticipated asymptotic formula for holds for all with at most exceptions.
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