AI 中文总结
本文针对Erdős关于素数立方和表示数的猜想给出无条件证明，利用Hecke等分布定理完成k=3的情况，再结合Green–Tao–Ziegler定理得到k=4时的下界，论证无法推广到k=4的情况。
AI 中文摘要
设\ntitle_cn\nF_k(n)为允许重复时，将n表示为素数的k次方和的无序表示数，即n=p₁ᵏ+p₂ᵏ+…+p_kᵏ。Erdős提出limsupₙ→∞ F₃(n)=∞，但其证明似乎未发表，本文给出完整的无条件证明。主要输入是CM费马三次型的经典Hecke等分布定理，其余论证使用算术级数中素数的标准估计和基础计数。k=3的论证无法推广到k=4的情况，但通过将Green–Tao–Ziegler定理应用于可容许二元四次恒等式产生的线性型，证明了limsupₙ→∞ F₄(n)≥2。
英文摘要
Let $F_k(n)$ be the number of unordered representations \[ n=p_1^k+p_2^k+\cdots +p_k^k \] by primes, with repetitions allowed. Erdős stated that $\limsup_{n\to\infty} F_3(n)=\infty$, but his proof appears not to have been published. A complete unconditional proof is given. The principal input is the classical Hecke equidistribution theorem for the CM Fermat cubic; the rest of the argument uses standard estimates for primes in arithmetic progressions and elementary counting. The argument used for \(k=3\) does not extend to the case \(k=4\). Nevertheless, by applying the Green--Tao--Ziegler theorem to the linear forms arising from an admissible binary quartic identity, \(\limsup_{n\to\infty}F_4(n)\ge2\) is shown.
