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    "title": "Chebyshev: rigorous bounds on how often primes appear",
    "summary": "The1852 memoir replaces prime-table conjectures with quantitative bounds on weighted prime counts and uses them to prove Bertrand’s interval claim beyond the finite range covered by tables.\n\nA table can show many examples of primes between a number and roughly twice that number, but it cannot rule out an exception farther away. Chebyshev links the logarithm of a factorial to sums over primes and prime powers, then bounds those sums. That turns a pattern in a table into a statement about every sufficiently large input. He closes the remaining finite cases using prime tables, preserving Bertrand’s role in posing the conjecture. The bounds also constrain how many primes occur, while remaining weaker than the exact asymptotic relation proved later in the prime number theorem.",
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        "description": "The target is quantitative upper/lower weighted-prime and interval-count bounds and the original Bertrand proof for integer a>3: a prime lies strictly between a and 2 a−2. Original 381–382 treats a>160 analytically and smaller admissible integers by prime tables; finite table checks alone do not prove the infinite range. Theta sums natural logarithms over primes up to x, distinct from the auxiliary prime-power sum. These bounds do not prove pi(x)~x/log(x), a zero-free zeta theorem or general primes in every shorter interval. Later series applications impose positivity/monotonicity conditions and are not advertised here as universal convergence criteria. no independent full proof replay. Presentation 1850 and publication 1852 are distinguished without duplicate counting.",
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      "statement": "# Chebyshev: rigorous bounds on how often primes appear\n\nThe1852 memoir replaces prime-table conjectures with quantitative bounds on weighted prime counts and uses them to prove Bertrand’s interval claim beyond the finite range covered by tables.\n\nA table can show many examples of primes between a number and roughly twice that number, but it cannot rule out an exception farther away. Chebyshev links the logarithm of a factorial to sums over primes and prime powers, then bounds those sums. That turns a pattern in a table into a statement about every sufficiently large input. He closes the remaining finite cases using prime tables, preserving Bertrand’s role in posing the conjecture. The bounds also constrain how many primes occur, while remaining weaker than the exact asymptotic relation proved later in the prime number theorem.\n\n## Scientific scope\n\nThe target is quantitative upper/lower weighted-prime and interval-count bounds and the original Bertrand proof for integer a>3: a prime lies strictly between a and 2 a−2. Original 381–382 treats a>160 analytically and smaller admissible integers by prime tables; finite table checks alone do not prove the infinite range. Theta sums natural logarithms over primes up to x, distinct from the auxiliary prime-power sum. These bounds do not prove pi(x)~x/log(x), a zero-free zeta theorem or general primes in every shorter interval. Later series applications impose positivity/monotonicity conditions and are not advertised here as universal convergence criteria. no independent full proof replay. Presentation 1850 and publication 1852 are distinguished without duplicate counting.\n\n## People and systems\n\nTchebichef (Chebyshev)\n\n\n\n## Original sources\n\n- [Tchebichef, Mémoire sur les nombres premiers (1852), original366–390](https://www.numdam.org/item/JMPA_1852_1_17__366_0.pdf)\n- [Numdam/BnF Mathdoc catalogue of the1852 Tchebichef memoir](https://www.numdam.org/item/JMPA_1852_1_17__366_0/)\n",
      "summary": "The1852 memoir replaces prime-table conjectures with quantitative bounds on weighted prime counts and uses them to prove Bertrand’s interval claim beyond the finite range covered by tables.\n\nA table can show many examples of primes between a number and roughly twice that number, but it cannot rule out an exception farther away. Chebyshev links the logarithm of a factorial to sums over primes and prime powers, then bounds those sums. That turns a pattern in a table into a statement about every sufficiently large input. He closes the remaining finite cases using prime tables, preserving Bertrand’s role in posing the conjecture. The bounds also constrain how many primes occur, while remaining weaker than the exact asymptotic relation proved later in the prime number theorem.",
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