Problem detail · source-aware
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Can $S$-decoding polynomials modulo a product of $k$ primes be built with only $k+1$ nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any constant $s$ there is an $s$-server private information retrieval protocol with communication $\exp(O((\log n)^{1/s}(\log\log n)^{1-1/s}))$ on an $n$-bit database, where previous constructions at that communication needed $2^{O(s)}$ servers.
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Stated in the abstract itself rather than buried in an acknowledgement: the main result for constant $s$ and its proof were discovered in a GPT-5.5 Pro conversation prompted by the authors.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
arXiv preprint; the authors also empirically validate the construction and make the result unconditional for all $s \le 15$. Not yet peer-reviewed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.