Elsholtz 2001 inverse goldbach problem

../

corollary_p2: There is no three-summand sumset A+B+C, each summand with at least two elements, that coincides with the set of primes for all sufficiently large elements.

theorem_p1: If two sets of positive integers, each with at least two elements, have a sumset that agrees with the primes beyond some point, then for large x both counting functions lie, up to constant factors, between x^{1/2}/(log x)^5 and x^{1/2}(log x)^4.

---

Elsholtz, Christian, The inverse Goldbach problem. Mathematika 48 (2001), 151-158, DOI 10.1112/S0025579300014406. The copy read for this card is the author's own version from the author's page (https://www.math.tugraz.at/~elsholtz/WWW/papers/papers.html, read 2026-10-02), which states no copyright notice, license or download terms, and that version prints only "Submission September 7, 2000 (this version includes galley corrections). Appeared in Mathematika 2001." and no notice; the term is unstated.

For sets $\mathcal{A},\mathcal{B}$ of positive integers with $|\mathcal{A}|,|\mathcal{B}|\ge2$ whose sumset $\mathcal{A}+\mathcal{B}$ coincides with the primes beyond some $x_0$, the paper's unnumbered Theorem (pp. 1-2) proves $x^{1/2}(\log x)^{-5}\ll A(x)\ll x^{1/2}(\log x)^4$ for all sufficiently large $x$, and the same for $B(x)$, improving the bounds of Hornfeck, of Hofmann and Wolke, and of the author's earlier note (p. 2). The proof (Section 2, pp. 2-7; for the Theorem pp. 3-7) lets Montgomery's sieve on $\mathcal{A}$ and Gallagher's larger sieve on $\mathcal{B}$ share the residue classes modulo each prime, first proving the weaker Proposition (p. 4), $x^{1/2-\varepsilon}\ll A(x),B(x)\ll x^{1/2+\varepsilon}$, then iterating. With a special case of a theorem of Pomerance, Sárközy and Stewart (Lemma 1, p. 2), the lower bound gives the Corollary (p. 2): no sumset of three sets of at least two elements each coincides with the primes for all sufficiently large elements. The paper calls Ostmann's two-summand question still open (p. 1).

Source: https://www.math.tugraz.at/~elsholtz/WWW/papers/papers.html.

Results. Labels and pages are those of the author's version (pp. 1-8); the Theorem and the Corollary are unnumbered.

• Theorem (pp. 1-2; proof pp. 3-7): if $\mathcal{P}'=\mathcal{A}+\mathcal{B}$ with $|\mathcal{A}|,|\mathcal{B}|\ge2$ and $\mathcal{P}'$ coinciding with the primes beyond $x_0$, then $x^{1/2}(\log x)^{-5}\ll A(x)\ll x^{1/2}(\log x)^4$ for $x\ge x_1$, and the same for $B(x)$.
• Corollary (p. 2; proof p. 2): there are no sets $\mathcal{A},\mathcal{B},\mathcal{C}$ with $|\mathcal{A}|,|\mathcal{B}|,|\mathcal{C}|\ge2$ whose sumset coincides with the primes for sufficiently large elements.

Read status. Claims checked for both results: the statements were read clause by clause on the page images of the author's version; the proofs were read but not checked step by step.

Bears on. #431: the Theorem (pp. 1-2) applies to any two infinite sets of positive integers the problem asks for and shows that both counting functions would lie between $x^{1/2}(\log x)^{-5}$ and $x^{1/2}(\log x)^4$ up to constants; it does not decide whether such sets exist, a question the paper calls still open (p. 1). The Corollary (p. 2) settles only the three-summand analogue, in the negative.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.