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2026
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| Online Access: | https://doi.org/10.5281/zenodo.20258227 |
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| _version_ | 1866901675093000192 |
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| author | Zhang, Jian |
| author_facet | Zhang, Jian |
| contents | <p>Cracking the Hidden Algebraic Structure in Ramanujan's π Formula: A₁ is Not an Integer, But a 12th-Degree Algebraic Number</p> <p>For a century, the origin of the enormous integers (such as 13591409) in Ramanujan's π formulas has remained a mystery. This preprint thoroughly corrects a previous erroneous claim: for the first level (m=1) of Ramanujan-type series, the parameter A₁ is not equal to the ground-state integer A₀, but is a 12th-degree algebraic number.</p> <p>Through systematic high-precision algebraic verification across all nine class-number-one discriminants, we confirm:</p> <p>For b = 4, 7, 8, 11, 19, 43, 67, 163, A₁ is a 12th-degree irreducible algebraic number with Galois group C₁₂.</p> <p>b = 3 (j=0) degenerates due to the presence of 6th roots of unity; the degree of its A₁ remains undetermined.</p> <p>The minimal polynomial of ε₁ = A₁ − 13591409 for b = 163 is given (degree 12, with large but exact coefficients).</p> <p>Explicit degree-12 minimal polynomials are provided for b = 67 and b = 8.</p> <p>Simplified degree-12 polynomials for the associated constants τ (Crystal Time Factor) and k (Structural Constant) are also provided, with remarkably small coefficients.</p> <p>This work does not rely on any unconventional axioms. All results can be independently reproduced using the attached Python verification script (verify_short.py).</p> <p>Files included:</p> <p>short_correction.pdf (the paper)</p> <p>verify_short.py (verification script)</p> <p>supplementary_polynomials.txt (supplementary polynomials and numerical values)</p> <p>Related resources:</p> <p>Independent Correction Note (DOI: 10.5281/zenodo.20248731)</p> <p>CT-3 Compendium (DOI: 10.5281/zenodo.20065612)</p> <p>Keywords: Ramanujan-type series, series for π, algebraic number, 12th-degree minimal polynomial, Chudnovsky formula, Crystal Time Factor, Structural Constant</p> <p>License: CC BY 4.0</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20258227 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Degree-12 Algebraic Correction to the Level-1__Parameter__of Ramanujan-Type Series for 1_π.pdf Zhang, Jian <p>Cracking the Hidden Algebraic Structure in Ramanujan's π Formula: A₁ is Not an Integer, But a 12th-Degree Algebraic Number</p> <p>For a century, the origin of the enormous integers (such as 13591409) in Ramanujan's π formulas has remained a mystery. This preprint thoroughly corrects a previous erroneous claim: for the first level (m=1) of Ramanujan-type series, the parameter A₁ is not equal to the ground-state integer A₀, but is a 12th-degree algebraic number.</p> <p>Through systematic high-precision algebraic verification across all nine class-number-one discriminants, we confirm:</p> <p>For b = 4, 7, 8, 11, 19, 43, 67, 163, A₁ is a 12th-degree irreducible algebraic number with Galois group C₁₂.</p> <p>b = 3 (j=0) degenerates due to the presence of 6th roots of unity; the degree of its A₁ remains undetermined.</p> <p>The minimal polynomial of ε₁ = A₁ − 13591409 for b = 163 is given (degree 12, with large but exact coefficients).</p> <p>Explicit degree-12 minimal polynomials are provided for b = 67 and b = 8.</p> <p>Simplified degree-12 polynomials for the associated constants τ (Crystal Time Factor) and k (Structural Constant) are also provided, with remarkably small coefficients.</p> <p>This work does not rely on any unconventional axioms. All results can be independently reproduced using the attached Python verification script (verify_short.py).</p> <p>Files included:</p> <p>short_correction.pdf (the paper)</p> <p>verify_short.py (verification script)</p> <p>supplementary_polynomials.txt (supplementary polynomials and numerical values)</p> <p>Related resources:</p> <p>Independent Correction Note (DOI: 10.5281/zenodo.20248731)</p> <p>CT-3 Compendium (DOI: 10.5281/zenodo.20065612)</p> <p>Keywords: Ramanujan-type series, series for π, algebraic number, 12th-degree minimal polynomial, Chudnovsky formula, Crystal Time Factor, Structural Constant</p> <p>License: CC BY 4.0</p> |
| title | A Degree-12 Algebraic Correction to the Level-1__Parameter__of Ramanujan-Type Series for 1_π.pdf |
| url | https://doi.org/10.5281/zenodo.20258227 |