Divisibility of Central Binomial Coefficients by the Partition Function: A Computational Study and Evidence for Finiteness
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2025
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| author | Naladiga Venkat, Arvind |
| author_facet | Naladiga Venkat, Arvind |
| contents | <p>We investigate the divisibility condition p(n) | $C(2n,n)$, where p(n) is the integer partition function and $C(2n,n)$ is the central binomial coefficient. Through a novel computational approach utilizing partial factorization up to $2n$, we determine that this divisibility occurs for exactly twelve values in the range $n <= 20000$: namely n in ${0, 1, 2, 4, 5, 6, 8, 11, 14, 17, 18, 26}$. Our algorithm leverages two rigorous obstructions: (1) the prime bound property, efficiently verified through residual computation without full factorization, and (2) q-adic valuation constraints via Legendre's formula for all primes q. Exhaustive computation finds no solutions beyond n = 26, with zero candidates passing both obstruction tests for any $n > 100$. Based on this computational evidence and theoretical analysis, we conjecture that these twelve values constitute the complete solution set.</p> <p>This record archives the complete Python source code, computational logs, and research paper associated with this project.</p> <h3>Key Features</h3> <ul> <li>Efficient Divisibility Checking: Implements a fast algorithm to test the condition $p(n) \mid \binom{2n}{n}$.</li> <li>Large-Scale Search: Capable of analyzing the problem for a large range of $n$ (tested up to $n=20,000$).</li> <li>Resumable Computation: Generates and stores partition data in chunked `.pkl` files, allowing computation to be stopped and resumed.</li> <li>Congruence Analysis: Includes functions to compute the exact value of $\binom{2n}{n} \pmod{p(n)}$ for cases where all prime factors of $p(n)$ are known.</li> <li>Pattern Discovery: Can be used to search for other patterns and congruences in the remainders.</li> </ul> <h3>Included Files</h3> <ul> <li> `<em>venkat-2025-partition-binomial-divisibility.pdf</em>`: The accompanying research paper.</li> <li> `<em>analyze_partitions.py</em>`: The core Python script to run the computation and analysis.</li> <li> `<em>requirements.txt</em>`: A list of the Python dependencies required to run the script.</li> <li> `<em>computation_log.txt</em>`: The log file capturing the time taken to run the script.<br><br></li> </ul> <h3>Licensing</h3> <p>This project uses a dual-license model:</p> <ul> <li>Source Code (.py file): MIT License</li> <li>All Other Files (Paper, documentation, results): Creative Commons Attribution 4.0 International (CC BY 4.0)</li> </ul> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17162713 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Divisibility of Central Binomial Coefficients by the Partition Function: A Computational Study and Evidence for Finiteness Naladiga Venkat, Arvind Mathematics central binomial coefficient partition function divisibility Legendre's formula p-adic valuations prime bound partial factorization Chinese remainder theorem computational number theory obstruction method experimental mathematics MSC2020:11B65 MSC2020: 11A25 MSC2020: 05A17 MSC2020: 11Y11 integer partitions binomial modulo m <p>We investigate the divisibility condition p(n) | $C(2n,n)$, where p(n) is the integer partition function and $C(2n,n)$ is the central binomial coefficient. Through a novel computational approach utilizing partial factorization up to $2n$, we determine that this divisibility occurs for exactly twelve values in the range $n <= 20000$: namely n in ${0, 1, 2, 4, 5, 6, 8, 11, 14, 17, 18, 26}$. Our algorithm leverages two rigorous obstructions: (1) the prime bound property, efficiently verified through residual computation without full factorization, and (2) q-adic valuation constraints via Legendre's formula for all primes q. Exhaustive computation finds no solutions beyond n = 26, with zero candidates passing both obstruction tests for any $n > 100$. Based on this computational evidence and theoretical analysis, we conjecture that these twelve values constitute the complete solution set.</p> <p>This record archives the complete Python source code, computational logs, and research paper associated with this project.</p> <h3>Key Features</h3> <ul> <li>Efficient Divisibility Checking: Implements a fast algorithm to test the condition $p(n) \mid \binom{2n}{n}$.</li> <li>Large-Scale Search: Capable of analyzing the problem for a large range of $n$ (tested up to $n=20,000$).</li> <li>Resumable Computation: Generates and stores partition data in chunked `.pkl` files, allowing computation to be stopped and resumed.</li> <li>Congruence Analysis: Includes functions to compute the exact value of $\binom{2n}{n} \pmod{p(n)}$ for cases where all prime factors of $p(n)$ are known.</li> <li>Pattern Discovery: Can be used to search for other patterns and congruences in the remainders.</li> </ul> <h3>Included Files</h3> <ul> <li> `<em>venkat-2025-partition-binomial-divisibility.pdf</em>`: The accompanying research paper.</li> <li> `<em>analyze_partitions.py</em>`: The core Python script to run the computation and analysis.</li> <li> `<em>requirements.txt</em>`: A list of the Python dependencies required to run the script.</li> <li> `<em>computation_log.txt</em>`: The log file capturing the time taken to run the script.<br><br></li> </ul> <h3>Licensing</h3> <p>This project uses a dual-license model:</p> <ul> <li>Source Code (.py file): MIT License</li> <li>All Other Files (Paper, documentation, results): Creative Commons Attribution 4.0 International (CC BY 4.0)</li> </ul> |
| title | Divisibility of Central Binomial Coefficients by the Partition Function: A Computational Study and Evidence for Finiteness |
| topic | Mathematics central binomial coefficient partition function divisibility Legendre's formula p-adic valuations prime bound partial factorization Chinese remainder theorem computational number theory obstruction method experimental mathematics MSC2020:11B65 MSC2020: 11A25 MSC2020: 05A17 MSC2020: 11Y11 integer partitions binomial modulo m |
| url | https://doi.org/10.5281/zenodo.17162713 |