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    "summary": "Miller gives deterministic compositeness-witness algorithms and establishes polynomial-bit-length primality testing conditional on the stated Extended Riemann Hypothesis, alongside number-theoretic functional reductions under declared assumptions.\n\nTo decide whether a huge integer is composite, one can look for a short arithmetic witness instead of trying every possible divisor. Miller combines modular-power checks with bounds on how small such witnesses must be. A fast deterministic guarantee follows under his stated Extended Riemann Hypothesis; a separate unconditional algorithm has a weaker cost. The paper also distinguishes checking a proposed factorization from actually finding it. Later randomized tests and unconditional deterministic tests remove different limitations and remain separate contributions.",
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        "description": "Exact 1976 J. Comput. Deterministic polynomial bit-length primality result assumes source Appendix ERH for stated Dirichlet-character L-functions; unconditional test has a different non-polynomial-bit-length bound. Perfect-power/root/divisor checks and small witnesses remain. Source abstract shorthand and refined theorem bit costs differ; damaged exponent/log glyphs not certified, no guessed exact complexity pasted in. Eulerphi/Carmichael/factorization equivalences and period oracle reductions retain ERH and source reduction/size conditions; recognizable factorization graph is not efficient general factor generation. No resolution of ERH, Pvs NP or factoring. Tiny own modular checks illustrate witnesses, not inherited analytic lemmas/globalproof. No independent scientific certification.",
        "domain": "Computational number theory"
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      "statement": "# Miller: conditional efficient deterministic primality testing\n\nMiller gives deterministic compositeness-witness algorithms and establishes polynomial-bit-length primality testing conditional on the stated Extended Riemann Hypothesis, alongside number-theoretic functional reductions under declared assumptions.\n\nTo decide whether a huge integer is composite, one can look for a short arithmetic witness instead of trying every possible divisor. Miller combines modular-power checks with bounds on how small such witnesses must be. A fast deterministic guarantee follows under his stated Extended Riemann Hypothesis; a separate unconditional algorithm has a weaker cost. The paper also distinguishes checking a proposed factorization from actually finding it. Later randomized tests and unconditional deterministic tests remove different limitations and remain separate contributions.\n\n## Scientific scope\n\nExact 1976 J. Comput. Deterministic polynomial bit-length primality result assumes source Appendix ERH for stated Dirichlet-character L-functions; unconditional test has a different non-polynomial-bit-length bound. Perfect-power/root/divisor checks and small witnesses remain. Source abstract shorthand and refined theorem bit costs differ; damaged exponent/log glyphs not certified, no guessed exact complexity pasted in. Eulerphi/Carmichael/factorization equivalences and period oracle reductions retain ERH and source reduction/size conditions; recognizable factorization graph is not efficient general factor generation. No resolution of ERH, Pvs NP or factoring. Tiny own modular checks illustrate witnesses, not inherited analytic lemmas/globalproof. No independent scientific certification.\n\n## People and systems\n\nGary L. Miller\n\n\n\n## Original sources\n\n- [Miller: conditional efficient deterministic primality testing; exact original journal technical text](https://www.cs.cmu.edu/~glmiller/Publications/Papers/Mi76.pdf)\n- [Publisher-deposited exact 1976 journal identity](https://api.crossref.org/works/10.1016%2FS0022-0000%2876%2980043-8)\n",
      "summary": "Miller gives deterministic compositeness-witness algorithms and establishes polynomial-bit-length primality testing conditional on the stated Extended Riemann Hypothesis, alongside number-theoretic functional reductions under declared assumptions.\n\nTo decide whether a huge integer is composite, one can look for a short arithmetic witness instead of trying every possible divisor. Miller combines modular-power checks with bounds on how small such witnesses must be. A fast deterministic guarantee follows under his stated Extended Riemann Hypothesis; a separate unconditional algorithm has a weaker cost. The paper also distinguishes checking a proposed factorization from actually finding it. Later randomized tests and unconditional deterministic tests remove different limitations and remain separate contributions.",
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