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    "title": "Copeland–Erdős normality of prime and sufficiently dense integer concatenations",
    "summary": "For a strictly increasing positive-integer sequence whose counting function eventually exceeds N^theta for every theta<1, concatenating its base-b representations gives a b-normal real. This proves the prime-concatenation example and the sums-of-two-squares example using the stated counting estimates.",
    "boundary": "Fixed-base normality of the corresponding constructed real, not absolute normality of the same decimal in every base or algorithmic randomness. The polynomial-value and simultaneous-base assertions at the end are conjectures, not proved here. Not absolute normality or algorithmic randomness. Finite leading digits do not alter normality. Besicovitch and Borel counting ingredients and Champernowne earlier construction keep credit; later polynomial and simultaneous-base paragraphs are explicitly conjectural in this source.",
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    "year": 1946,
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        "title": "Note on normal numbers, Bulletin AMS52(1946)857–860",
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        "description": "Fixed-base normality of the corresponding constructed real, not absolute normality of the same decimal in every base or algorithmic randomness. The polynomial-value and simultaneous-base assertions at the end are conjectures, not proved here. Not absolute normality or algorithmic randomness. Finite leading digits do not alter normality. Besicovitch and Borel counting ingredients and Champernowne earlier construction keep credit; later polynomial and simultaneous-base paragraphs are explicitly conjectural in this source.",
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      "statement": "# Copeland–Erdős normality of prime and sufficiently dense integer concatenations\n\nFor a strictly increasing positive-integer sequence whose counting function eventually exceeds N^theta for every theta<1, concatenating its base-b representations gives a b-normal real. This proves the prime-concatenation example and the sums-of-two-squares example using the stated counting estimates.\n\n## Scientific scope\n\nFixed-base normality of the corresponding constructed real, not absolute normality of the same decimal in every base or algorithmic randomness. The polynomial-value and simultaneous-base assertions at the end are conjectures, not proved here. Not absolute normality or algorithmic randomness. Finite leading digits do not alter normality. Besicovitch and Borel counting ingredients and Champernowne earlier construction keep credit; later polynomial and simultaneous-base paragraphs are explicitly conjectural in this source.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nArthur H. Copeland; Paul Erdős\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Note on normal numbers, Bulletin AMS52(1946)857–860](https://users.renyi.hu/~p_erdos/1946-01.pdf)\n",
      "summary": "For a strictly increasing positive-integer sequence whose counting function eventually exceeds N^theta for every theta<1, concatenating its base-b representations gives a b-normal real. This proves the prime-concatenation example and the sums-of-two-squares example using the stated counting estimates.",
      "title": "Copeland–Erdős normality of prime and sufficiently dense integer concatenations"
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    "id": "copeland-erdos-dense-integer-concatenation-normality-1946",
    "entry_id": "copeland-erdos-dense-integer-concatenation-normality-1946",
    "revision": 1,
    "title": "Copeland–Erdős normality of prime and sufficiently dense integer concatenations",
    "context": "Number theory and probability",
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    "what_changed": "For a strictly increasing positive-integer sequence whose counting function eventually exceeds N^theta for every theta<1, concatenating its base-b representations gives a b-normal real. This proves the prime-concatenation example and the sums-of-two-squares example using the stated counting estimates.",
    "scope": "For a strictly increasing positive-integer sequence whose counting function eventually exceeds N^theta for every theta<1, concatenating its base-b representations gives a b-normal real. This proves the prime-concatenation example and the sums-of-two-squares example using the stated counting estimates. Fixed-base normality of the corresponding constructed real, not absolute normality of the same decimal in every base or algorithmic randomness. The polynomial-value and simultaneous-base assertions at the end are conjectures, not proved here. Not absolute normality or algorithmic randomness. Finite leading digits do not alter normality. Besicovitch and Borel counting ingredients and Champernowne earlier construction keep credit; later polynomial and simultaneous-base paragraphs are explicitly conjectural in this source.",
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