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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.19599355 |
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| _version_ | 1866901683524599808 |
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| author | Helmdach, Emma |
| author_facet | Helmdach, Emma |
| contents | <div class="Y3BBE"><strong class="Yjhzub">Abstract:</strong></div> <div class="Y3BBE">This paper presents a novel approach to explaining the validity of Fermat's Last Theorem and Euler's sum of powers conjecture through the internal architecture of numbers. The author utilizes a fundamental arithmetic principle: any natural number raised to the power k (N^k) can be represented as a sum of N consecutive odd numbers.</div> <div class="Y3BBE">By analyzing these sequences, the paper demonstrates a critical distinction between the second power and all higher powers:</div> <ol class="IaGLZe VimKh"> <li class="dF3vjf"><span class="T286Pc">For squares (k=2), the sequences are nested and continuous, starting from 1, which allows for the existence of Pythagorean triples (A^2 + B^2 = C^2).</span></li> <li class="dF3vjf"><span class="T286Pc">For higher powers (k > 2), a "structural gap" emerges as the starting odd number of each sequence shifts forward at an accelerating rate, defined by the formula X = N^(k-1) - (N-1).</span></li> </ol> <div class="Y3BBE">The author argues that the impossibility of decomposing a power into the sum of only two others for k > 2 is caused by a divergence in "nominal weight" (density) of the odd numbers. Even if the quantity of numbers in a sum is correct, their collective arithmetic mass from the beginning of the series cannot match the density required for a target block of a higher order.</div> <div class="Y3BBE">Furthermore, this model provides an arithmetic justification for Euler's conjecture, suggesting that at least k summands are required to "stitch" the structural gap inherent in the k-th degree. This theoretical framework serves as a foundation for the author's practical discovery of parametric series for "quadruples" of cubes (A^3 = B^3 + C^3 + D^3).</div> <div class="Fsg96"> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19599355 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Explanation of why powers of a number greater than two cannot be decomposed into the sum of two terms of the same power Helmdach, Emma Fermat's Last Theorem Euler's conjecture Odd numbers, Arithmetic Sum of powers Diophantine equations <div class="Y3BBE"><strong class="Yjhzub">Abstract:</strong></div> <div class="Y3BBE">This paper presents a novel approach to explaining the validity of Fermat's Last Theorem and Euler's sum of powers conjecture through the internal architecture of numbers. The author utilizes a fundamental arithmetic principle: any natural number raised to the power k (N^k) can be represented as a sum of N consecutive odd numbers.</div> <div class="Y3BBE">By analyzing these sequences, the paper demonstrates a critical distinction between the second power and all higher powers:</div> <ol class="IaGLZe VimKh"> <li class="dF3vjf"><span class="T286Pc">For squares (k=2), the sequences are nested and continuous, starting from 1, which allows for the existence of Pythagorean triples (A^2 + B^2 = C^2).</span></li> <li class="dF3vjf"><span class="T286Pc">For higher powers (k > 2), a "structural gap" emerges as the starting odd number of each sequence shifts forward at an accelerating rate, defined by the formula X = N^(k-1) - (N-1).</span></li> </ol> <div class="Y3BBE">The author argues that the impossibility of decomposing a power into the sum of only two others for k > 2 is caused by a divergence in "nominal weight" (density) of the odd numbers. Even if the quantity of numbers in a sum is correct, their collective arithmetic mass from the beginning of the series cannot match the density required for a target block of a higher order.</div> <div class="Y3BBE">Furthermore, this model provides an arithmetic justification for Euler's conjecture, suggesting that at least k summands are required to "stitch" the structural gap inherent in the k-th degree. This theoretical framework serves as a foundation for the author's practical discovery of parametric series for "quadruples" of cubes (A^3 = B^3 + C^3 + D^3).</div> <div class="Fsg96"> </div> |
| title | Explanation of why powers of a number greater than two cannot be decomposed into the sum of two terms of the same power |
| topic | Fermat's Last Theorem Euler's conjecture Odd numbers, Arithmetic Sum of powers Diophantine equations |
| url | https://doi.org/10.5281/zenodo.19599355 |