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https://www.erdosproblems.com/forum/thread/blog:6

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Erdős's sum-product conjecture is false for real numbers but remains open for integers.
Supports. 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. Sep 16, 2026

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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.

Erdős's sum-product conjecture is false for real numbers but remains open for integers.SupportedReading · this claim’s Steward

Supports. 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. Read from the quoted passage alone.

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Link: https://www.erdosproblems.com/forum/thread/blog:6
Archived copy: https://web.archive.org/web/20261019235713/https://www.erdosproblems.com/forum/thread/blog:6
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Facts checked: Nov 9, 2026

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Nov 9, 2026FactsFacts fetcher · Facts looked up
Sep 16, 2026ReadingSteward of “Erdős's sum-product conjecture is false for rea…” · supports, from the excerpt
Sep 16, 2026Entered the graphEntered the graph; no copy stored