Claims

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1937_04_01_erdos: Erdős proves that infinitely many n have more than n^(c/(log log n)^2) representations as a sum of two squares of primes, sharpened to n^(c/log log n) in Part II, so limsup f_2(n) is infinite; refereed.

1965_01_01_erdos: Erdős's 1965 statement that he could also prove limsup f_3(n) = infinity for sums of three cubes of primes, in unpublished work that seemed to need special properties of primes; no proof by him has appeared.

2026_08_17_kitamura: An independent Lean 4 proof that the number of representations of n as a sum of three cubes of primes is unbounded, the case k = 3; registered as the formal-conjectures proof of that variant, not built here.

2026_08_18_wang_zhang: A preprint proving unconditionally that some integers have arbitrarily many representations as sums of three cubes of primes, the case k = 3, through Hecke equidistribution for the Fermat cubic; not refereed.

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