Geometric Obstructions to Sums of Higher Powers with Remarks Related to Beal's Conjecture
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901255343833088 |
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| author | Gordanpour, Niloufar |
| author_facet | Gordanpour, Niloufar |
| contents | <p>This note presents a heuristic and geometric perspective on Beal’s Conjecture, an open problem in number theory asserting that any solution to </p> <p>a^x + b^y = c^z \quad (x,y,z > 2)</p> <p>must involve integers sharing a common prime divisor. The work introduces a five-principle framework: <br>1. uneven expansion of higher powers, <br>2. necessity of a common divisor for monomial collapse, <br>3. explosion of intermediate terms, <br>4. incompatibility of geometric growth rates for coprime bases, <br>5. lattice obstructions from incompatible sublattices. </p> <p>While not a formal proof, this approach provides a structural and geometric explanation for the rigidity of higher-power sums and clarifies why the quadratic (Pythagorean) case is exceptional.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18471093 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometric Obstructions to Sums of Higher Powers with Remarks Related to Beal's Conjecture Gordanpour, Niloufar Beal's Conjecture Exponential Diophantine equations Number theory Heuristic approach Geometric interpretation of powers Lattice obstruction Common prime divisor <p>This note presents a heuristic and geometric perspective on Beal’s Conjecture, an open problem in number theory asserting that any solution to </p> <p>a^x + b^y = c^z \quad (x,y,z > 2)</p> <p>must involve integers sharing a common prime divisor. The work introduces a five-principle framework: <br>1. uneven expansion of higher powers, <br>2. necessity of a common divisor for monomial collapse, <br>3. explosion of intermediate terms, <br>4. incompatibility of geometric growth rates for coprime bases, <br>5. lattice obstructions from incompatible sublattices. </p> <p>While not a formal proof, this approach provides a structural and geometric explanation for the rigidity of higher-power sums and clarifies why the quadratic (Pythagorean) case is exceptional.</p> |
| title | Geometric Obstructions to Sums of Higher Powers with Remarks Related to Beal's Conjecture |
| topic | Beal's Conjecture Exponential Diophantine equations Number theory Heuristic approach Geometric interpretation of powers Lattice obstruction Common prime divisor |
| url | https://doi.org/10.5281/zenodo.18471093 |