Syracuse
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Fichiers liés à publication SSRN Syracuse Conjecture Algorithmic and Modular Analysis of Increasing and decreasing segments https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5105381
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Understanding the Calculation Rule Multiply any odd number 𝑛 by 3, add 1, and repeatedly divide by 2 until the result is odd. Continue applying the rule to successive odd numbers until reaching 1. Setting Up the Algorithm Use a programming language or tool of your choice (e.g., Python, MATLAB, or Excel) to implement the calculation rule. Include a module-calculating function to determine the congruence of each number modulo 8, 16, or 32 et 64. Segment Identification Define increasing and decreasing segments based on the sequence behavior: Increasing segment: 𝑛𝑖 + 1 > ni Decreasing segment: 𝑛𝑖 + 1 < 𝑛𝑖 Ensure the transition between segments is identified using modular patterns (e.g., numbers congruent to 3mod16). Replication of Fifty Syracuse Sequences Generate 50 Syracuse sequences starting from various initial numbers. For each sequence: Identify and label segments as increasing or decreasing. Record modular patterns for each transition. Probabilistic Analysis Compute the theoretical frequencies of decreasing segments using modular periodicities. Compare empirical frequencies from the sequences to the theoretical predictions. Verification Cross-validate the results by applying the calculation rule to additional numbers or datasets. Use provided supplementary files (e.g., "Fifty Syracuse Sequences," "Periodicities," and "Theoretical Frequency") for comparison.