Erdős's sum-product conjecture is false for real numbers but remains open for integers.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The sum-product conjecture, first stated by Erdős in 1976 and usually credited to Erdős and Szemerédi, holds that for any finite set A of numbers either the sumset A+A or the product set AA must have nearly quadratic size, at least |A|^{2-o(1)}. Erdős asked it for the integers, the reals and the complex numbers, with the integers his main interest. For fifty years the question was open in every setting, with lower bounds stuck just above the exponent 4/3, and the conjecture was widely believed.
In May 2026 Thomas Bloom, Will Sawin, Carl Schildkraut and Dmitrii Zhelezov posted a disproof over the reals (arXiv:2605.28781). They construct arbitrarily large finite sets of algebraic integers, taken from totally real number fields whose degree grows like the logarithm of the set's size, for which both the sumset and the product set are at most |A|^{2-c} for an absolute constant c > 0. The idea is a high-dimensional version of the Balog–Wooley example: a box in the additive lattice of the ring of integers multiplied by a box in the unit lattice, with class-field towers supplying fields of bounded root discriminant. The paper was inspired by OpenAI's counterexample to the unit distance conjecture a week earlier. It is a preprint and has not yet been refereed, but it has been expounded by one of its authors, re-derived independently by AI systems using partly different constructions, and used as a template in further refutations by other authors; no objection to it has surfaced. On that basis the conjecture over the reals should now be regarded as false.
The construction says nothing about the integers, or about any number field of bounded degree, because the counterexamples require the degree to grow with the set. The authors state that the conjecture may still be true in the original integer setting, and Bloom has written that whether integer counterexamples exist remains wide open and would likely need much deeper number theory. The conjecture over the integers is therefore still open, with the best known lower bound around |A|^{4/3+c}. The claim as a whole is accurate on the current record; what would raise it from well supported to established is refereed publication or a machine-checked formalization of the real-number disproof, and what would falsify its second half is a proof or disproof of the integer case.
Full reasoning: the evidence and decisions behind this verdict
The claim is a two-part status statement, and both parts were checked against the primary source rather than the news report the claim was extracted from.
First half (false over the reals). The paper by Bloom, Sawin, Schildkraut and Zhelezov (arxiv.org/abs/2605.28781, full text at arxiv.org/html/2605.28781v1) states Theorem 1.1: there is an absolute constant c > 0 and arbitrarily large finite A ⊂ ℝ with max(|A+A|,|AA|) ≤ |A|^{2-c}. It is deduced from Theorem 1.2, which for infinitely many degrees d gives totally real fields K and sets A ⊂ O_K with X^d ≤ |A| ≤ (CX)^d, |A+A| ≤ C^d|A| and |AA| ≤ 2^{-d}|A|^2; an embedding of K into ℝ and the choice X = C^{1/ε} gives Corollary 1.3 and hence Theorem 1.1. Section 5 sketches an explicit c ≥ 0.00000087. The introduction is careful about scope and matches the claim's framing exactly. Independent corroboration: Huang's paper (arxiv.org/html/2607.20525v1) reports that a GPT-5.5 Pro agent, with no access to the published proof, produced correct disproofs in 7 of 8 trials, several avoiding units altogether, and records that Alpöge announced an independent AI proof; a follow-up by Roche-Newton, Schildkraut and Warren (arxiv.org/abs/2606.24583) and Pohoata's Elekes–Rónyai counterexample reuse the construction; Schildkraut presented the result at Stanford; the paper thanks Fox, Peluse and Dembner for reading it. No published objection was found. Under the mathematics standard, a recent unrefereed proof, however well received, sits at supported rather than verified; that is the reason for the status, not any doubt found in the argument. The seeded subclaim the sum-product conjecture over the reals carries this evidence and should come out contradicted.
Second half (open over the integers). The paper says the construction's fields have degree tending to infinity, "and so (1.1) may still be true in number fields of bounded degree (and, in particular, the original setting of ℤ)." Bloom's expository post of 31 May 2026 (www.erdosproblems.com/forum/thread/blog:6, read through a search excerpt because the site refused direct fetching) says it "remains wide open whether counterexamples exist to the sum-product conjecture in the integers." The Erdős Problems entry for Problem 52 lists the best lower bound as |A|^{1270/951-o(1)} (Bloom) and the Erdős–Szemerédi upper bound |A|^2 exp(-c log|A|/log log|A|). A search for any integer resolution since May 2026 found none. The subclaim the sum-product conjecture over the integers must stay unresolved for this half to hold; either a proof or a disproof there would falsify it.
Instances. Five sources, all affirming: the paper, Bloom's blog, Huang's paper, the Roche-Newton–Schildkraut–Warren paper (the last two affirm the real-number half explicitly and are recorded at reduced confidence), and the Quanta feature of 3 August 2026, which restates the paper faithfully with the hedge "a version of". The distribution is lopsided because the discourse is; no counterweight exists to record.
Credence 0.94 for the conjunction: about 0.95 that the real disproof stands as written (the argument is short, elementary relative to the unit distance construction, and has been reproduced by independent routes), and near certainty that the integer case is open as of this pass. What would change the verdict: an error found in the construction (unlikely given the independent re-derivations), refereed publication or a checked formalization (upgrade to verified), or any resolution of the integer problem (the claim would become false).
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- contradictsthis argues against the parentsteward instructions →Every finite set A of real numbers satisfies max(|A+A|, |AA|) ≥ |A|^{2-o(1)}. ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →Every finite set A of integers satisfies max(|A+A|, |AA|) ≥ |A|^{2-o(1)}. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The mathematicians found a set of real numbers for which both the sum and the product grow more slowly than expected. The conjecture for integers remains open.
About the "sum-product" conjecture: either the sum or product of a set of numbers must grow quickly.
The source's own evidence bears what it asserts. A news feature that reports the result in one paragraph, hedging that the group "disproved a version of" the conjecture; it rests entirely on the Bloom, Sawin, Schildkraut and Zhelezov paper and adds no evidence of its own. Its summary matches what that paper claims.
In this paper we prove that the sum-product conjecture (1.1) is false over the reals, by constructing arbitrarily large counterexamples in totally real algebraic number fields of large degree. The degree of these fields tends to infinity as the sets grow (like ≍ log n for a counterexample of size n), and so (1.1) may still be true in number fields of bounded degree (and, in particular, the original setting of ℤ).
The primary source: the paper disproving the conjecture over the reals by constructing sets of algebraic integers in high-degree totally real number fields, and stating in its introduction that the integer case is untouched by the construction.
The source's own evidence bears what it asserts. This is the paper that establishes the result. Its introduction is careful about what the construction does and does not do: the number fields have degree growing like the logarithm of the set size, so the conjecture may still hold in fields of bounded degree and in the integers. The proof is a preprint, not yet refereed, but has been independently expounded and reused. Worth reading closely: It is the proof itself; any future doubt about the disproof is settled by reading Sections 3 and 4 (the additive and unit lattices and the construction). The quoted passage was not found in the stored copy of this source.
Finally, note that these sets $A$ get arbitrarily large, but only exist in number fields of degree $d\to \infty$. In particular they are far from subsets of the integers, which is the main setting which concerned Erdős (although he did also ask it for the reals). It remains wide open whether counterexamples exist to the sum-product conjecture in the integers, and I expect any proof or disproof to require much deeper number theory than is being used here.
Bloom's expository blog post sketching the disproof of the sum-product conjecture over the reals and the unit distance counterexample, and stating that the integer case remains wide open. Read via search excerpt only; the page could not be fetched directly.
The source's own evidence bears what it asserts. The author's own exposition of the proof, stating plainly that the integer case remains wide open. The page could not be fetched during this pass; the passage was read from a search excerpt.
Inspired by the unit-distance counterexample, Bloom, Sawin, Schildkraut, and Zhelezov soon made a second breakthrough: a human disproof of the Erdős–Szemerédi sum-product conjecture over ℝ.
A paper reporting that a GPT-5.5 Pro agent independently produced correct disproofs of the conjecture over the reals in 7 of 8 trials, some by constructions different from the published one. It affirms the real-number half of the claim and notes Erdős's emphasis on the integer case without stating its status.
The sum-product problem has seen some remarkable progress recently, with the refutation of the sum-product conjecture over the reals by Bloom, Sawin, Schildkraut and Zhelezov [3].
A follow-up paper (Roche-Newton, Schildkraut, Warren) that builds on the construction to refute further conjectures in the area; it affirms the refutation over the reals as established. Read via search excerpt.
How these sources relate
- https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/ draws its statement from https://arxiv.org/abs/2605.28781, faithfully. The article's statement is a report of the paper's result; the paper is not cited by name but is unambiguously identified (four mathematicians including Bloom, real numbers, technique related to the unit distance counterexample). The summary matches the paper's own framing, including the caveat about integers; if anything it is hedged ("a version of") rather than strengthened.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.