Ostmann’s inverse Goldbach conjecture

2 · Going further

For two sets $A,B\subseteq {\mathbb{Z}}_{\ge 0}$, their sumset is defined by $A+B=\{a+b:a\in A,b\in B\}$. The question is whether a set $S$ differing from the set $\mathcal{P}$ of primes by only finitely many elements can be written as $S=A+B$, with $|A|\ge 2$ and $|B|\ge 2$. The main result rules out every such representation.

The condition on $S$ permits both adding and removing finitely many elements. The required equality is exact: every sum must belong to $S$, and every element of $S$ must occur as a sum. This is the precise sense in which the manuscript’s title describes the primes as additively indecomposable.

The repository summary and the manuscript abstract present this statement as a resolution of Ostmann’s inverse Goldbach conjecture. However, the supplied abstract describes neither the method nor the steps of the argument, so it does not support an explanation of the proof strategy here. The main result has a Lean formalization, but the supplied text does not detail its technical scope. No companion manuscript is supplied.

Useful notions

• Prime numbers
• Nonnegative integers
• Sets and membership
• Sets of all possible sums
• Finite modification of a set

What was known before / what is new

Limits and assumptions

The statement concerns sets of nonnegative integers, each containing at least two elements, and requires exact equality with the set of all their sums. It covers only sets differing from the primes by finitely many elements; it gives no conclusion for infinite modifications or other number domains. A Lean formalization of the main result is reported, but its detailed scope and the mathematical argument are absent from the supplied text. The source mentions no remaining open question.

Mini glossary

Prime number

—

An integer greater than 1 whose only positive divisors are 1 and itself.

Sumset

—

The set $A+B$ consists of every number $a+b$ with $a\in A$ and $b\in B$.

Finite modification

—

A change to a set made by adding or removing finitely many elements.

Additive indecomposability

—

In this statement, the impossibility of writing the set in question as $A+B$ with two sets of nonnegative integers containing at least two elements each.

Keywords : Prime numbers · Additive indecomposability · Ostmann’s conjecture

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family