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Statement

Satz II (p. 668, quoted). "In jedem Intervall $(0, x)$ liegen mehr als $\beta x$ Zahlen, welche als Summe von einer Primzahl und einer Potenz von $a$ darstellbar sind. Hier ist $a$ eine gegebene ganze Zahl und $\beta$ eine positive Konstante, welche nur von $a$ abhängt."

In the corpus's words: for a given integer $a$ there is a constant $\beta>0$, depending only on $a$, such that every interval $(0,x)$ contains more than $\beta x$ integers of the form $p+a^n$ with $p$ prime.

The paper says only that $a$ is a given integer. Its proof needs $a\ge2$: it counts the powers of $a$ up to $x$ as $N(x)=[\log x/\log a]$ (11, p. 672), which requires $a>1$. That count is the number of powers $a^n$ with $1\le n\le\log x/\log a$; whether $a^0=1$ is admitted does not matter, since the integers $p+1$ number at most $\pi(x)$ up to $x$. These remarks are observations of this page.

As with Satz I, "every interval $(0,x)$" is to be read for $x$ sufficiently large: the proof (p. 673) gives $\nu(2x)>2\beta x$ for large $x$, and footnote 1 (p. 668) defines positive density by $N(x)/x>\alpha$ for all sufficiently large $x$. The theorem is a positive lower density statement; the paper does not show that the density exists.

Source. N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 109 (1934), 668--678, doi:10.1007/BF01449161; Satz II on p. 668, its proof on pp. 671--673, and the convergence of the auxiliary series on pp. 673--678. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the proof were read clause by clause on the page images, including the convergence argument of pp. 673--678 and its Hilfssätze I and II (pp. 675--676); the cited estimates (Schnirelman's bound (5), the bounds (9) and (13)) were not re-derived. Nothing here is independently reviewed.

Proof pointer

Pp. 671--678. The inequality (1) of pp. 668--669 (see Satz I) is applied with the primes and the powers of $a$. Since a difference $a^i-a^j$ with $i\ne j$ determines $i$ and $j$, each $A_2(u,x)$ is $0$ or $1$ (p. 672). With Schnirelman's bound (5) and the inequality $f(v_1)f(v_2)\ge f(v_1v_2)$ for $f(u)=\prod_{q\mid u}(1+\frac1q)$, the correlation sum is reduced (pp. 672--673) to

$$\sum_{u=1}^{x}A_1(u,x)A_2(u,x)<c_8\,x\sum_{\substack{k=1\\(k,a)=1}}^{x}\frac{\mu(k)^2}{k\,l(k)},$$

where $l(k)$ is the multiplicative order of $a$ modulo $k$. The paper proves (pp. 673--678) that the series

$$\sum_{\substack{k=1\\(k,a)=1}}^{\infty}\frac{\mu(k)^2}{k\,l(k)}$$

converges, which gives (12), $\sum_uA_1A_2<c_9x$, and then (1) with (7) and (11) gives $\nu(2x)>2\beta x$ (p. 673). The display (7) is reprinted on p. 672 with $\log^2x$ in both bounds; the computation on p. 673 uses the Chebyshev bounds $c_2x/\log x<M(x)<c_3x/\log x$ of p. 670, which are what $M(x)=\pi(x)$ satisfies.

The convergence proof rewrites the series as $\sum_l\frac1l$ times a sum over the primitive squarefree divisors of $a^l-1$, bounds that by $\sigma(l)$, the sum of $\mu(d)^2/d$ over divisors $d$ of $a^l-1$ with $\varphi(d)\equiv0\pmod l$ (p. 674), and proves that $\sum_l\sigma(l)/l$ converges. Writing $l=l_1l_2^2$ with $l_1$ squarefree, the terms with $l_2\ge l_1$ are handled by $\sigma(l)<c_{10}\log l$ (14, p. 675). For the terms with $l_1>l_2$ the paper proves two auxiliary results. Hilfssatz I (p. 675): if $\pi(x,k)$, the number of primes $kz+1$ up to $x$, is at least $2$, then $\pi(x,k)<b_1x/(k^{1/3}\log x)$ with $b_1$ absolute. Hilfssatz II (p. 676): the reciprocals of the first $y$ primes $\equiv1\pmod k$ sum to less than $b_2\log\log y/k^{1/3}$ for every $y\ge3$, with $b_2$ absolute. These give $\sigma(l)<b_6\,l^{\varepsilon_1}l_1^{-1/3}\sum_i\Theta_i(l_1)/i!$ (p. 678), where $\Theta_i(l_1)$ counts the factorizations of $l_1$ into $i$ factors greater than $1$, and the second part of the series is then bounded by $b_6\zeta(2+2\varepsilon_2)e^{\zeta(1+\varepsilon_2)}$.

Dependencies

Inequality (1) of the same paper (pp. 668--669); Schnirelman's generalization of Brun's results, quoted as (5) (p. 670); Chebyshev's bounds for $\pi(x)$ (7); the bound $f(u)<c_{13}\log\log u$ from the known estimates for Euler's function (13, p. 675); the bound on the number of prime factors used on p. 678.

Bears on

• Problem 244: for an integer $C\ge2$ the problem's integers $p+\lfloor C^k\rfloor$ are the integers $p+C^k$, so Satz II with $a=C$ gives them positive lower density; it says nothing about non-integer $C$. The problem's claim page for this paper records this partial result.
• Problem 851: with $a=2$, Satz II says that the integers $2^k+p$ have positive lower density, the case of one prime divisor with a positive constant in place of the problem's $1-\epsilon$; it does not give the density bound the problem asks for.
• Problem 16: with $a=2$, Satz II gives positive lower density to the integers $2^k+p$, and so, since at most $\log x/\log2$ of them up to $x$ have $p=2$, to the odd integers of that form (an observation of this page). The problem's claim page cites the theorem for this fact; the theorem does not decide the problem.
• Problem 205: the problem lists the paper as a reference. With $a=2$, Satz II gives a positive lower density of integers $2^k+m$ with $m$ prime, so $\Omega(m)=1$; the problem asks about all sufficiently large integers, and the theorem does not bear on its answer.