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A Constructive Proof of the Infinitude of Mersenne Primes via Bit Templates in Lucas–Lehmer Sequences

Bibliographic Details
Title: A Constructive Proof of the Infinitude of Mersenne Primes via Bit Templates in Lucas–Lehmer Sequences
Authors: Okuda, Tetsuya
Publisher Information: Zenodo
Publication Year: 2025
Collection: Zenodo
Subject Terms: Mersenne primes, Lucas-Lehmer test, constructive proof, bit templates, Hamming weight, primality testing, binary automaton
Description: We present a constructive and deterministic proof for the infinitude of Mersenne primes. By analyzing the bit-level structure of the Lucas–Lehmer sequence, we identify a recurring “collapse template” of Hamming weights that guarantees successful primality testing. We show that such a pattern can always be constructed and detected algorithmically, thus proving the existence of infinitely many Mersenne primes.
Document Type: Thesis
Language: unknown
Relation: https://zenodo.org/records/16255004; oai:zenodo.org:16255004; https://doi.org/10.5281/zenodo.16255004
DOI: 10.5281/zenodo.16255004
Availability: https://doi.org/10.5281/zenodo.16255004
https://zenodo.org/records/16255004
Rights: Creative Commons Attribution 4.0 International ; cc-by-4.0 ; https://creativecommons.org/licenses/by/4.0/legalcode
Accession Number: edsbas.8397A03E
Database: BASE
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  – Url: https://doi.org/10.5281/zenodo.16255004#
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  Data: A Constructive Proof of the Infinitude of Mersenne Primes via Bit Templates in Lucas–Lehmer Sequences
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Okuda%2C+Tetsuya%22">Okuda, Tetsuya</searchLink>
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  Data: Zenodo
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  Group: Date
  Data: 2025
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  Data: <searchLink fieldCode="DE" term="%22Mersenne+primes%22">Mersenne primes</searchLink><br /><searchLink fieldCode="DE" term="%22Lucas-Lehmer+test%22">Lucas-Lehmer test</searchLink><br /><searchLink fieldCode="DE" term="%22constructive+proof%22">constructive proof</searchLink><br /><searchLink fieldCode="DE" term="%22bit+templates%22">bit templates</searchLink><br /><searchLink fieldCode="DE" term="%22Hamming+weight%22">Hamming weight</searchLink><br /><searchLink fieldCode="DE" term="%22primality+testing%22">primality testing</searchLink><br /><searchLink fieldCode="DE" term="%22binary+automaton%22">binary automaton</searchLink>
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  Data: We present a constructive and deterministic proof for the infinitude of Mersenne primes. By analyzing the bit-level structure of the Lucas–Lehmer sequence, we identify a recurring “collapse template” of Hamming weights that guarantees successful primality testing. We show that such a pattern can always be constructed and detected algorithmically, thus proving the existence of infinitely many Mersenne primes.
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  Data: https://zenodo.org/records/16255004; oai:zenodo.org:16255004; https://doi.org/10.5281/zenodo.16255004
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  Data: 10.5281/zenodo.16255004
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      – Type: doi
        Value: 10.5281/zenodo.16255004
    Languages:
      – Text: unknown
    Subjects:
      – SubjectFull: Mersenne primes
        Type: general
      – SubjectFull: Lucas-Lehmer test
        Type: general
      – SubjectFull: constructive proof
        Type: general
      – SubjectFull: bit templates
        Type: general
      – SubjectFull: Hamming weight
        Type: general
      – SubjectFull: primality testing
        Type: general
      – SubjectFull: binary automaton
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      – TitleFull: A Constructive Proof of the Infinitude of Mersenne Primes via Bit Templates in Lucas–Lehmer Sequences
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              M: 01
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              Y: 2025
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