华林-哥德巴赫问题: 两个平方, 两个立方和两个四次方

令$P_r$表示素因子不超过 $r$ 的殆素数, 按重数计. 作者证明了对于充分大的偶数 $N$, 方程 $$N=x^2+p_1^2+p_2^3+p_3^3+p_4^4+p_5^4$$ 有解, 其中 $x$ 是殆素数 $P_6$, $p_j\,(j=1, \cdots, 5)$ 是素数.

Waring-Goldbach Problem: Two Squares, Two Cubes and Two Biquadrates

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every sufficiently large even integer N, theequation N = x2 + p21 + p32 + p33 + p44 + p45 is solvable with x being an almost-prime P6 and the pj (j = 1, · · · , 5) are primes.