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Statement

Setting (p. 113). $f(n)$ is the number of solutions of $2^k+p=n$ with $p$ prime, as for Theorem 1. The paper recalls (p. 113) that Romanoff proved

$$\limsup\frac1x\sum_{n=1}^{x}f^2(n)<\infty, \qquad (1)$$

from which, with Cauchy--Schwarz and the count of more than $c_3x$ pairs $(k,p)$ with $2^k+p\le x$, the integers $2^k+p$ have positive density.

Theorem 2 (p. 113). For every $k$,

$$\limsup\frac1x\sum_{n=1}^{x}f^k(n)<\infty. \qquad (3)$$

Here $k$ is the exponent of the moment, not the exponent of the power of $2$; the case $k=2$ is Romanoff's (1).

Proof pointer

Pp. 115--119. Writing $\varphi(x;i_1,\ldots,i_k)$ for the number of solutions of $p_{i_1}+2^{i_1}=\cdots=p_{i_k}+2^{i_k}$ in primes below $x$, inequality (10) reduces the moment to $k^k[\sum\varphi(x;i_1,\ldots,i_k)+x]$ over distinct $i$'s with $2^i\le x$. Brun's sieve, in the form of Erdős's 1937 paper, bounds each $\varphi$ by $x(\log x)^{-k}$ times a product over the primes dividing the differences $2^{i_u}-2^{i_v}$ (11); the arithmetic--geometric mean inequality and (13) reduce the theorem to the convergence of $\sum_d B^{v(d)}/(d\,l_2(d))$ (14), where $l_2(d)$ is the order of $2$ modulo $d$ and $v(d)$ the number of distinct prime factors of $d$. As in Erdős and Turán's proof of Romanoff's $\sum 1/(d\,l_2(d))<\infty$, the $d$ are split by whether $l_2(d)<(\log d)^{c_{13}}$; the first class is sparse by a count of integers composed of the prime factors of $2^k-1$, $k\le(\log x)^{c_{13}}$ (16)--(19), and the second is handled by partial summation (20)--(22).

Read depth

Claims checked: the statement and recalled result (1) on p. 113 were read on the page images, and the proof on pp. 115--119 was followed in outline; its estimates were not re-derived. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Brun's method as in P. Erdős, Proc. Cambridge Philos. Soc. 33 (1937), 6--12 (its footnote 7), and the Erdős--Turán proof of Romanoff's series bound, cited through Landau's Cambridge tract (its footnotes 1 and 8). Romanoff's paper has its own source card.

Source. P. Erdős, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math. 2 (1950), fasc. 8, 113--123; the edition read is named on the source card.

Bears on

No problem page directly. The theorem bounds $f$ on average; it gives no pointwise bound of the kind Problem 236 asks for, which the paper poses as a conjecture on p. 115.