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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 $S₃\left(X\right)={\sum }_{1\le m₁,m₂,m₃\le X}\tau (m₁²+m₂²+m₃²)$. We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_{\epsilon }\left(X^{8/3+\epsilon }\right)$ obtained by Guo and Zhai (2012).

Let $R_s\left(n\right)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s=3$ or 4, it is established that the anticipated asymptotic formula for $R_s\left(n\right)$ holds for all $n\le X$ with at most $O\left(X^{(9-2s)/8+\epsilon }\right)$ exceptions.