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Source. Christian Elsholtz, The inverse Goldbach problem, Mathematika 48 (2001), 151-158, read in the author's version identified on the source card; labels and pages are that version's (pp. 1-8). The Corollary is unnumbered.

Statement

The paper states (p. 2), as "Corollary (Solution of the inverse ternary Goldbach problem)": "There do not exist sets of integers $\mathcal{A},\mathcal{B}$, and $\mathcal{C}$ with $|\mathcal{A}|,|\mathcal{B}|,|\mathcal{C}|\geq 2$, and a set $\mathcal{P}'$ which coincides with the set of primes $\mathcal{P}$ for sufficiently large elements such that $\mathcal{A}+\mathcal{B}+\mathcal{C}=\mathcal{P}'$ holds."

In words: the primes, changed in finitely many elements, are never a sumset of three sets of at least two elements each. The paper adds after the proof that the same conclusion holds for more than three summands (p. 2).

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 2 of the author's version, and the short proof on p. 2 was read.

Proof pointer

Section 2, p. 2. Grouping two of the three summands, each summand is one term of a two-set decomposition, so the lower bound of the Theorem gives $A(x),B(x),C(x)\gg x^{1/2-\varepsilon}$. Lemma 1 (p. 2), a special case of Theorem 3 of Pomerance, Sárközy and Stewart, then gives an element $a_1+b+c\ge x^{0.4}$ of the sumset, with $a_1\ge x^{0.4}$, divisible by a prime $p\le x^{1/3+\varepsilon}$, which cannot lie in $\mathcal{P}'$ for large $x$.

Dependencies

The Theorem (lower bound) and Lemma 1 (p. 2), which the paper cites from Pomerance, Sárközy and Stewart.

Bears on

• Problem 431: the problem asks about two summands; the corollary settles only the three-summand analogue, with no infiniteness assumption, and says nothing about whether two infinite sets can have such a sumset.