Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer Peter W. Shor

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 Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer 1997 SIAM Journal on Computing / arXiv 10.1137/S0097539795293172 quant-ph/9508027 shor1997-factoring shor1997-factoring.txt, the project's staged copy 521342bd9582cdca50d1970a641023c2feb1e03f arxiv-open quantum-computing

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 0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. mathematical-proof Quotation withheld because this record is not cleared for redistribution. A THESIS'S VALUE IS ITS ECONOMY, AND THE ECONOMY IS STATED AS THE REASON: Church's thesis GREATLY SIMPLIFIES THE STUDY OF COMPUTATION, SINCE IT REDUCES THE POTENTIAL FIELD OF STUDY FROM AN INFINITE NUMBER OF POTENTIAL COMPUTING DEVICES TO TURING MACHINES. What is claimed for it first is not its truth but what it lets one stop studying about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. FORMALISING A CLAIM WOULD MAKE IT WEAKER, AND THE MECHANISM IS GIVEN: the thesis IS NOT A MATHEMATICAL THEOREM, and to make it one WOULD REQUIRE A PRECISE MATHEMATICAL DESCRIPTION of a computing device - which WOULD LEAVE OPEN THE POSSIBILITY of some practical device NOT SATISFYING that description, and so WOULD MAKE THE RESULTING THEOREM WEAKER THAN THE ORIGINAL THESIS. A formalism can be escaped in a way an informal claim cannot about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A CATEGORY IS INTRODUCED EXPLICITLY AS A COMPROMISE BETWEEN TWO UNUSABLE GRAINS: the old distinction is MUCH TOO COARSE, the exact efficiency is TOO PRECISE A QUANTITY TO WORK WITH EASILY, and what the field uses is THE GENERALLY ACCEPTED COMPROMISE BETWEEN COARSENESS AND PRECISION - polynomial versus superpolynomial scaling. The class P is presented as a negotiated grain, not a natural joint about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. THE CLASSIFICATION IS SHOWN TO REQUIRE A FURTHER THESIS BEFORE IT MEANS ANYTHING: FOR THIS CLASSIFICATION TO MAKE SENSE, WE NEED IT TO BE MACHINE-INDEPENDENT - we need to know that polynomial-time computability does not depend on the kind of device used - and that requirement IS a quantitative version of Church's thesis. A definition resting on an unproved claim about the world about this quote: RE-SLICED: the first cut ended "...quantitative version of Church", stopping INSIDE THE NAME. The cut-word guard did not catch it because the next character is a curly apostrophe, not a letter - so `Church` + `’s` reads as a clean word boundary to the check. A span must not end inside a person's name either, and no guard currently tests that. PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: `PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b` matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. ONE CLAIM CARRIES THREE NAMES ACROSS THREE LITERATURES, AND ONE OF THEM ONLY PARTLY: the same thesis has been CALLED THE STRONG CHURCH'S THESIS by one group, and MAKES UP HALF OF THE INVARIANCE THESIS of another author. Two vocabularies name it whole, a third contains it as a part - which is a partial overlap between naming systems, not a synonym about this quote: EXTENDED BACKWARDS: the first cut began "which Vergis et al. [1986] have called..." and never named the thesis it was about. A tag asserting that one claim carries three names has to contain the claim. Overlaps the span of the preceding mechanism, which is allowed and normal. PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A READER-FACING SUBSTITUTION IS OFFERED AND JUSTIFIED BY MUTUAL SIMULATION: readers NOT COMFORTABLE WITH Turing machines MAY THINK INSTEAD OF digital computers with linearly growing memory, BECAUSE THESE TWO CLASSES CAN EFFICIENTLY SIMULATE EACH OTHER. One object is licensed to stand in for another, and the licence is an equivalence about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. THE THESIS'S ESCAPE CLAUSES ARE LOCATED IN TWO SPECIFIC WORDS AND NAMED AS SUCH: THERE ARE TWO ESCAPE CLAUSES IN THE ABOVE THESIS - ONE OF THESE IS THE WORD PHYSICAL, and machine models that violate the thesis HAVE BEEN RULED OUT BY SOME REASON FOR WHY THEY ARE NOT PHYSICAL, that is, why they could not be built and made to work. A single underdefined word is doing the excluding about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. AND THE SECOND ESCAPE CLAUSE IS A WORD WHOSE MEANING IS SAID TO BE INCOMPLETELY SPECIFIED: THE OTHER ESCAPE CLAUSE IS THE WORD RESOURCES, THE MEANING OF WHICH IS NOT COMPLETELY SPECIFIED - after which the resources are enumerated for digital computers, and MORE are said to be pertinent to analog computation. The thesis's content varies with a list nobody has closed about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A NEW RESOURCE IS ADDED TO THE ACCOUNTING AND THE ADDITION IS ARGUED FOR: for quantum computation, IN ADDITION TO SPACE AND TIME, THERE IS ALSO A THIRD POTENTIALLY IMPORTANT RESOURCE, PRECISION - and the argument is that the changes a quantum computer makes CAN CLEARLY NOT BE PERFECTLY ACCURATE but MUST CONTAIN SOME SMALL AMOUNT OF INHERENT IMPRECISION. The bookkeeping is extended because the physics forces a third column about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. WHAT SETS THE LIMIT IS ATTRIBUTED TO MATERIALS RATHER THAN TO PHYSICAL LAW, WITH THE HEDGE STATED: AS FAR AS WE KNOW, what precision is possible IS DICTATED NOT BY FUNDAMENTAL PHYSICAL LAWS BUT BY THE PROPERTIES OF THE MATERIALS AND THE ARCHITECTURE with which the computer is built - and IT IS CURRENTLY NOT CLEAR which architectures will give high precision about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A MODELLING CHOICE IS DEFENDED AGAINST A TRUE DESCRIPTION, AND THE GROUND IS USEFULNESS: treating precision as a large constant - EVEN THOUGH IT IS ALMOST CERTAIN TO BE CONSTANT FOR ANY GIVEN MACHINE - is compared to treating a digital computer as a finite automaton, of which the verdict is THIS VIEW IS TECHNICALLY CORRECT; HOWEVER, IT IS NOT PARTICULARLY USEFUL. A description can be accurate and the wrong one to adopt about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A KEY WORD IS FOUND SUBSTITUTED IN ANOTHER AUTHOR'S STATEMENT OF THE SAME THESIS, AND THE SUBSTITUTION IS DIAGNOSED: in one restatement THE WORD PHYSICAL IS REPLACED WITH THE WORD REASONABLE - and IT IS DIFFICULT TO IMAGINE ANY DEFINITION OF REASONABLE IN THIS CONTEXT WHICH DOES NOT MEAN PHYSICALLY REALIZABLE. One word swapped for another in the same claim, and the author argues they must come to the same thing about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. A THESIS IS BELIEVED ON THE STRENGTH OF FAILED REFUTATIONS, AND THE FAILURES ARE THEN SHOWN TO SHARE A PROPERTY: computer scientists became convinced THROUGH THE FAILURE OF ALL PROPOSED COUNTER-EXAMPLES - but MOST OF THESE HAVE BEEN BASED ON THE LAWS OF CLASSICAL MECHANICS, HOWEVER THE UNIVERSE IS IN REALITY QUANTUM MECHANICAL. The induction's instances share exactly the property the new case lacks about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. PRIORITY IS ASSIGNED BY THE EXPLICITNESS OF A QUESTION RATHER THAN BY THE IDEA, AND ONE AUTHOR IS CREDITED FOR WHAT HE DID NOT ASK: one researcher DID NOT ASK whether quantum mechanics conferred extra power AND YET showed enough to make later work possible; another RAISED THE POSSIBILITY, THUS IMPLICITLY ASKING the converse question; and a third WAS THE FIRST TO ASK THIS QUESTION EXPLICITLY about this quote: PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. precision Quotation withheld because this record is not cleared for redistribution. quantum computer Quotation withheld because this record is not cleared for redistribution. Church’s thesis Quotation withheld because this record is not cleared for redistribution. physical Quotation withheld because this record is not cleared for redistribution. reasonable Quotation withheld because this record is not cleared for redistribution. THREE PAPERS, THREE DIFFERENT VERDICTS ON WHETHER A FORMALISM CAN BE SHOWN TO CAPTURE THE INFORMAL NOTION IT REPLACES - and no type in the vocabulary can express a disagreement about what KIND OF CLAIM something is. This paper says the identification CANNOT BE MADE A THEOREM WITHOUT WEAKENING IT: a precise mathematical description of a computing device would leave open the possibility of a practical device that did not satisfy it, so the theorem would be weaker than the thesis it replaced. turing1936-computable-numbers argues it instead, enumerating three kinds of argument; tkacik2016-information-living reports a theorem that did make a vague notion precise. 'same-as' and 'distinct-from' are both false here: the three are not making one claim or two, they are ruling differently on whether the identification is the sort of thing a proof can settle. All eight edge types assert that a relation holds between two concepts. Recorded per record, each row quoting only its own paper; the cross-paper half is this paraphrase, checkable by opening the other two., and the third case for an earlier note of ours - which by the rule that one is an observation and three make a proposal is the point at which the gap should be put to Shir. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. differs in: THE AXIS IS WHETHER QUANTUM MECHANICS IS TAKEN INTO ACCOUNT. 'A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most A POLYNOMIAL FACTOR. This may not be true when quantum mechanics is taken into cons[IDERATION].' The thesis is not shown false - it is shown to have a physical precondition nobody had needed to state, and the polynomial factor is what fails. A 'contradicts' alone records a dispute with a computing thesis and drops the fact that the dispute is ABOUT WHICH PHYSICS THE MACHINE OBEYS. Quotation withheld because this record is not cleared for redistribution. physical reasonable strength of the relation: 0.8 · note: TWO PUBLISHED STATEMENTS OF ONE THESIS DIFFER IN ONE WORD, AND THE PAPER ARGUES THEY MUST MEAN THE SAME: in van Emde Boas [1990] '…is replaced with the word "reasonable." It is difficult…' A terminological argument inside a mathematics paper, about the load-bearing word of the thesis being tested. Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span says 'physical' was REPLACED by 'reasonable' in one statement of the thesis; same-as is loose. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. input size Quotation withheld because this record is not cleared for redistribution. number of steps Quotation withheld because this record is not cleared for redistribution. quantum order-finding algorithm Quotation withheld because this record is not cleared for redistribution. factoring integers Quotation withheld because this record is not cleared for redistribution. discrete logarithms Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1137/S0097539795293172.