Inverse Goldbach: why the primes cannot split into a sumset

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Family 013 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

The familiar Goldbach problem asks whether even integers can be expressed as sums of two primes. Ostmann’s inverse Goldbach problem reverses the structural question: can the primes themselves be obtained by adding elements from two simpler sets?

For sets $A,B$ of nonnegative integers, their sumset is

$$A+B=\{a+b:a\in A,\ b\in B\}.$$

The requirement is demanding. Every sufficiently large prime must occur, and every sufficiently large sum must be prime. It is not enough to represent many primes while also producing many composites.

Family 013 claims that such a decomposition is impossible when both summands contain at least two elements, even if one changes finitely many members of the prime set. This is Ostmann’s conjecture on the additive indecomposability of the primes.

The apparent simplicity hides a tension between local and global information. Avoiding composites forces restrictions modulo every prime. Covering all large primes forces the summands to be sufficiently large and sufficiently well coordinated. The manuscript aims to show that these demands cannot coexist.

2. Background and essential definitions

Let $\mathcal P$ be the positive primes and $\mathbb N_0=\{0,1,2,\ldots\}$. The symmetric difference

$$S\mathbin{\triangle}T=(S\setminus T)\cup(T\setminus S)$$

consists of elements belonging to exactly one of the sets. Saying that $(A+B)\triangle\mathcal P$ is finite means that the two sets agree beyond some threshold. This is eventual equality, not equality of densities or an approximation in proportion.

The condition on summand size removes trivial decompositions. For example, $\{0\}+\mathcal P=\mathcal P$, but one summand has only one element. Both finite and infinite sets are initially allowed as long as each has at least two elements.

Define the counting functions

$$A(x)=\#(A\cap[0,x]),\qquad B(x)=\#(B\cap[0,x]).$$

If every large prime at most $x$ must be a sum, then there must be enough pairs, so roughly

$$A(x)B(x)\gtrsim\frac{x}{\log x}.$$

Several pairs can represent the same prime, so this is only a necessary lower bound. An upper bound of order $x$ would still leave room between the two estimates.

For a prime $q$, the residue set $A\bmod q$ records which remainders occur. If sufficiently large $a\in A$ and $b\in B$ satisfy $a\equiv-b\pmod q$, then $a+b$ is divisible by $q$. Once the sum exceeds $q$, it is composite. Therefore the tails of $A$ and $-B$ must occupy disjoint residue sets modulo every $q$.

A sieve combines such local restrictions across many primes to bound the size of an integer set. An inverse sieve asks what structure a set must have if it nearly attains those bounds. Squares provide a model example: modulo an odd prime, they occupy about half the residue classes, yet there are about $\sqrt x$ squares below $x$.

3. Development history and previous partial results

Ostmann formulated the conjecture in his 1956 treatise on additive number theory. Laffer and Mann later showed that a hypothetical nontrivial eventual decomposition must have two infinite summands. This already rules out representing the primes as finitely many fixed translates of another set.

Work of Hornfeck, Pomerance–Sárközy–Stewart, Hofmann–Wolke, and Elsholtz progressively restricted possible decompositions. Elsholtz combined large and larger sieves to force both summands to have approximately square-root size, up to logarithmic factors, and ruled out three nontrivial summands. A ternary impossibility is not the binary result: adding a third set creates additional constraints.

Elsholtz and Harper sharpened the binary size restrictions. In their positive-integer formulation, a hypothetical decomposition forces each counting function between quantities of the approximate forms

$$\frac{\sqrt x}{\log x\log\log x} \quad\text{and}\quad \sqrt x\log\log x,$$

up to constants. Such bounds sharply narrow the possibilities without contradicting prime coverage.

Green and Harper’s inverse-sieve work proposed a structural route. Sets near the square-root bound and occupying about half the classes modulo primes should resemble quadratic images. A sufficiently strong classification would imply Ostmann’s conjecture. Their work established important partial results, not the required full inverse theorem.

Hanson proved that suitable sets have additive correlation and a substantial intersection with a quadratic image. More recent quadratic inverse-sieve work strengthened the intersection conclusions, as described in M1’s introduction. Large intersections still fall short of near-containment in one quadratic structure, especially when the quadratic may vary with the scale.

The manuscript claims a different final route: use the simultaneous residue restrictions together with coverage of every sufficiently large prime, without proving the general inverse-sieve classification. Its stated advance is a contradiction for the prime sumset problem itself.

4. What the new manuscripts claim

M1, Theorem 1.1, states that for every pair $A,B\subseteq\mathbb N_0$ with $|A|,|B|\ge2$,

$$(A+B)\mathbin{\triangle}\mathcal P \quad\text{is infinite}.$$

Equivalently, no finite modification of the set of primes can be written as a sumset of two nontrivial sets of nonnegative integers.

The conclusion contains two possible ways a candidate decomposition can fail: it can omit infinitely many primes, or contain infinitely many nonprimes, or both. In particular, if it contains every sufficiently large prime, then it must contain infinitely many composite numbers. The only nonprime nonnegative integers that are not composite are zero and one, so finite initial irregularities cannot account for that infinite failure.

No density or regularity assumptions on the summands are imposed. They need not be arithmetic progressions, polynomial images, computable sets, or sets with a known asymptotic counting formula.

The theorem does not say that two nontrivial sets cannot produce some primes, or even infinitely many primes. It says their full sumset cannot agree eventually with the full prime set. The exact coverage and exclusion conditions are both used in the manuscript.

5. Why the advance matters

Provisional importance: 4/5. Ostmann’s conjecture is a longstanding structural question about how an arithmetically defined set can arise from addition. Resolving it would show that the entire prime set has a rigidity invisible from its density alone.

The challenge is stronger than a simple parity obstruction. Choosing one summand even and the other odd can ensure odd sums, but oddness leaves many composites. One must simultaneously avoid divisibility by every prime while still reaching every large prime. Existing size bounds demonstrated the narrowness of that possibility; the claimed theorem rules it out completely.

The method would also connect local residue information, multiplicative character estimates, and global additive coverage. Its significance includes overcoming the need for a general structural classification that earlier conditional approaches sought. The score measures that mathematical importance if the claim is correct.

6. The main idea

Reduce to two infinite summands

Assume eventual equality. If $A$ were finite, two distinct elements of $A$ would require two fixed translates of the tail of $B$ to consist of primes. A standard upper-bound sieve limits such a set to order $x/(\log x)^2$. Finitely many translates of it cannot cover the order $x/\log x$ primes below $x$. M1 includes this short reduction before tackling infinite summands.

For the remaining case, set $D=-B$. At each prime $q$, sufficiently long tails of $A$ and $D$ occupy disjoint subsets of $\mathbb F_q$, the field of residues modulo $q$. Complete those subsets to a partition $S_q,S_q^c$. Collision estimates, which count how often two elements share a residue, show that on suitable averages these parts must be approximately balanced and the summands approximately uniform within them.

This creates many local partitions. The problem is to exploit their compatibility across primes without assuming they all have a simple algebraic form.

Eliminate persistent character patterns

A multiplicative character on $\mathbb F_q$ assigns roots of unity to nonzero residues and is multiplicative, with value zero at zero. The quadratic character distinguishes nonzero squares from nonsquares. Translating its argument produces a test centered at a chosen residue.

M1 first studies persistent correlations with translated quadratic characters. Moment estimates and Poisson summation are used to force independently chosen local centers into a common rational pattern. A quadratic large-sieve argument would then confine positive portions of the summands to a small family of quadratic structures, contradicting the required collision behavior.

Higher-order characters require further repeated Cauchy–Schwarz steps. Auxiliary anchor variables separate copied terms and control diagonal coincidences. The output is a mixed-character decorrelation estimate. This is a statement about the specific residue patterns forced by a hypothetical decomposition, not a general classification of all large sets with restricted residues.

Use prime coverage as an additional input

Avoiding composite sums supplied the residue partitions. Coverage of all large primes now supplies something different: their additive Fourier transforms cannot be too small on too much of the relevant prime average.

For a function $f$ on $\mathbb F_q$, its additive Fourier transform is

$$\widehat f(a)=\frac1q\sum_{x\in\mathbb F_q} f(x)e^{-2\pi i ax/q}.$$

It measures correlation with additive oscillations. M1 compares a nonnegative sumset statistic with an average over primes, deriving a lower bound that retains more coverage information than the preliminary estimate for $A(x)B(x)$.

This lower bound is central. A contradiction based only on sets avoiding certain residue classes would face the legitimate square-like examples that motivate the inverse-sieve problem. The full decomposition assumption contributes an extra positive statistic.

Force incompatible upper and lower bounds

Repeated transfers arrange terms in binary-tree patterns. A finite-field comparison uses the mixed-character decorrelation to control their correlations. Its hypotheses allow nonuniform pointwise values, which is necessary for the actual Fourier transforms of the residue indicators.

The arithmetic comparison then keeps track of repeated prime labels, coprimality restrictions, and the possible exceptional real character in prime-distribution estimates. In the final step, M1 averages over permutations of many bulk variables. Most pairs of arrangements overlap in too few components to maintain a large correlation. The remaining exceptional pairs and diagonal terms are bounded separately.

The resulting upper bound is claimed to contradict the positive lower bound inherited from prime coverage. The transfer depth is chosen large but fixed before the main scale grows. This is how the argument seeks to exclude every hypothetical eventual decomposition rather than merely improve its counting bounds.

M1 cites the zero-free theorem from family 003 only for an optional refinement in Remark 5.4. It explicitly states that the relevant proof uses its classical prime-distribution estimate and exceptional-character treatment instead. The indecomposability theorem is therefore not presented as depending on that new zero-free claim.

7. Scope, limitations, and open questions

The summands lie in the nonnegative integers, and each must contain at least two elements. The theorem concerns eventual equality after finitely many changes. It does not give a quantitative lower bound on how many composite sums a covering pair must produce.

A sumset can contain an interesting thin set of primes without being an eventual decomposition of all primes. Results and questions about such thin sumsets remain distinct. The theorem also does not resolve the ordinary binary Goldbach conjecture, whose target is the even integers and whose summands are primes.

The argument does not establish Green–Harper’s general inverse-sieve conjecture. A broad classification of sets near sieve bounds remains a separate structural problem. Other multiplicatively defined sets, restricted summand classes, and quantitative measures of failure require their own conclusions.

8. Sources and further reading

M1’s Theorem 1.1 states the eventual indecomposability result. Section 2 handles finite summands and residue restrictions; Sections 3–5 develop character decorrelation and the separate use of prime coverage; Sections 6–9 complete the tree comparison and contradiction.

The Elsholtz–Harper and Green–Harper papers linked in Section 3 explain the square-root barrier and the inverse-sieve approach. Optional background on sumsets and sieve theory may help place the problem within additive and analytic number theory.

Source manuscripts

• M1. OpenAI, The additive indecomposability of the primes (September 24, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.