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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.18712964 |
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Table of Contents:
- <p><span><span><span>This study is a continuation of previous research in </span><span> Geometric Decomposition of Mersenne Exponents: A Novel Additive Representation System </span><a href="https://zenodo.org/records/18256256"><span>(1).</span></a></span></span></p> <p> </p> <p><span><span><span>This paper presents an observed geometric pattern in the sequence of known Mersenne prime exponents (M1–M51). When plotted in a two-dimensional spiral arrangement based on index order and last digit, a remarkable structure emerges.</span></span></span></p> <p> </p> <p><span><span><span>Each Mersenne exponent corresponds to a unique point in the spiral</span></span></span></p> <ul> <li> <p><span>For every known Mersenne prime up to M51, there exist sets of 4 positive and 3 negative indices whose vector sum (in spiral coordinates) lands precisely on the target point</span></p> </li> <li> <p><span>These geometric visuals helps to identify patterns while the sums are revealing relationship</span></p> </li> <li> <p><span>The pattern reveals an unexpected additive structure linking earlier and later Mersenne primes</span></p> </li> </ul> <p><br><br></p> <p><span><span><span>"For any Mersenne exponent Mn (with a few exceptions), there exist sets of indices P and N, with ∣P∣=4, ∣N∣=3, such that in any ordinal embedding, the points corresponding to P form a triangle, the points corresponding to N form another triangle, and these triangles are balanced around Mn."</span></span></span></p> <p> </p> <p> </p>