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        "description": "Exact 1980 J. Composite-witness logic and at-most-quarter nonwitness bound yield <=4^-k false acceptance for each fixed composite under independent uniform random residues; this is not posterior primality probability or a deterministic prime certificate. Source inequality/cost glyphs damaged: inclusive quarter bound retained, exact strictness/step coefficient not certified. Arithmetic-operation model not silently unit bit cost; computation/randomness errors separate. Note eliminates gcds through square-root checks. Reported large/twin-prime searches are assertions, not twin-prime conjecture proof or guaranteed termination of all special-form searches; heuristic search costs remain heuristic. The complete proof and the source-reported computer experiments have not been independently reproduced here. Repeating one fixed base gains no independent-trial bound.",
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      "statement": "# Rabin: randomized primality testing with one-sided witness error\n\nRabin analyzes randomized compositeness-witness sampling: for each fixed composite, independent uniform trials give exponentially decreasing false acceptance, while a witness certifies compositeness; the journal paper includes implementation and reported experiments.\n\nA modular calculation can expose an integer as composite without finding all its factors. Rabin shows that witnesses are common enough that independent random choices rapidly reduce the chance of mistakenly accepting a composite. Finding a witness is decisive; passing the trials gives a probable-prime result, not a proof that the number is prime. The paper explains this distinction and reports implementations and large-number searches. Its later note simplifies the test using repeated squaring, while preserving the earlier witness theory and contributors.\n\n## Scientific scope\n\nExact 1980 J. Composite-witness logic and at-most-quarter nonwitness bound yield <=4^-k false acceptance for each fixed composite under independent uniform random residues; this is not posterior primality probability or a deterministic prime certificate. Source inequality/cost glyphs damaged: inclusive quarter bound retained, exact strictness/step coefficient not certified. Arithmetic-operation model not silently unit bit cost; computation/randomness errors separate. Note eliminates gcds through square-root checks. Reported large/twin-prime searches are assertions, not twin-prime conjecture proof or guaranteed termination of all special-form searches; heuristic search costs remain heuristic. The complete proof and the source-reported computer experiments have not been independently reproduced here. Repeating one fixed base gains no independent-trial bound.\n\n## People and systems\n\nMichael O. Rabin\n\n\n\n## Original sources\n\n- [Rabin: randomized primality testing with one-sided witness error; exact original journal technical text](https://www.cs.umd.edu/~gasarch/BLOGPAPERS/rabinprime.pdf)\n- [Publisher-deposited exact 1980 journal identity](https://api.crossref.org/works/10.1016%2F0022-314X%2880%2990084-0)\n",
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