Erdős's sum-product conjecture is false for real numbers but remains open for integers.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Read the extracting Quanta feature whole, then the primary source (Bloom–Sawin–Schildkraut–Zhelezov, arXiv:2605.28781, full HTML) and corroborating literature (Huang arXiv:2607.20525; Roche-Newton–Schildkraut–Warren arXiv:2606.24583; Bloom's Erdős Problems blog via search excerpt, since erdosproblems.com returned 403 to direct fetches, as did the Problem 52 page). Five web searches used. Structure: the graph held no node for the sum-product conjecture in any setting. Matcher returned "new" for both the real and integer propositions and suggested the two base-field variants be one node under the named-conjecture rule; I judged them distinct because the discourse now treats them as such (one refuted, one open, different truth values), which is the same-considerations test the mathematics skill makes decisive. Minted both as subclaims, with seeds and notes: the reals conjecture (02f443e5) as 'contradicts' (its truth would falsify the parent; seed 0.03), the integer conjecture (7d4df2ee) as 'requires' (a disproof would falsify the parent; a proof would too, which no relation type encodes; raised as a tool gap). The edge between the two propositions (integer case specifies the real case) belongs to the reals node's steward; noted in seed notes. Did not add a parent: my claim is a status claim about the propositions, which are its children; making the reals node a parent as well would create a cycle. No formalization attempted: a composite status statement ("remains open") is not a Prop; the reals node is the natural formalization target for its own steward. Instances: recorded four affirming instances (paper, Bloom blog, Huang, Roche-Newton et al.), the latter two at reduced confidence because they affirm only the real-number half. Quanta instance left as extracted (faithful, hedged). Provenance: readings on Quanta and the paper, a derives_from edge Quanta → paper (faithful), map written and marked immaterial; the quote check on the paper's instance reports not found only because the stored text is the abstract page while the passage is in the full-text HTML page I read. Importance kept at 0.45 (central conjecture, widely consulted, but undisputed status statement), contestation 0.35. Canonical form kept: it is the shortest neutral statement of what the discourse says, and "Erdős's" is defensible since the conjecture first appeared in Erdős 1976. Assessment: supported, confidence 0.78, credence 0.94, marginal yield 0.3. Status is supported rather than verified because the disproof is a four-month-old unrefereed preprint, per the mathematics standard; the evidence itself is strong and uniformly one-directional. No dependents exist, so no notification. No finding noted: the result is already what the discourse says.
Assessed Supported
verdict confidence 0.78 · credence 0.94
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.
Claim entered the graph