A Closed-Form Symbolic Generator: $A^n + B^n = C^n + D^n$, for $n = 2,3$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913909797027840 |
|---|---|
| author | Agbanwa, Jamal |
| author_facet | Agbanwa, Jamal |
| contents | We present a unified framework for constructing integer solutions to $A^{n} + B^{n} = C^{n} + D^{n}$ for $n=2,3$. For $n=2$, we derive explicit formulas for any solutions via differences of squares. For $n=3$, we introduce general formulas that include the Hardy-Ramanujan number 1729 for instance, we also construct a symbolic generator that produces infinitely many integer solutions to the Diophantine equation A^3 + B^3 = C^3 + D^3 . While the resulting formulas for $A,B,C,D$ from the symbolic generator developed do not span every single number expressible as a sum of two positive cubes in at least two distinct ways, our method provides a closed-form, algebraic parametrization in terms of a single variable, expressing each term as a radical-exponential function of an integer parameter $c_1$. The generator leverages nested radicals and exponents of algebraic numbers, $α, β$ derived from the recurrence structure of the Diophantine constraint. This work represents the first symbolic, recursive generator of its kind and offers a pathway toward approaching higher powers of this problem from a different lens. These methods exploit structural links between binomial expansions and Diophantine constraints, offering a foundation for extensions to higher powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19173 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Closed-Form Symbolic Generator: $A^n + B^n = C^n + D^n$, for $n = 2,3$ Agbanwa, Jamal General Mathematics We present a unified framework for constructing integer solutions to $A^{n} + B^{n} = C^{n} + D^{n}$ for $n=2,3$. For $n=2$, we derive explicit formulas for any solutions via differences of squares. For $n=3$, we introduce general formulas that include the Hardy-Ramanujan number 1729 for instance, we also construct a symbolic generator that produces infinitely many integer solutions to the Diophantine equation A^3 + B^3 = C^3 + D^3 . While the resulting formulas for $A,B,C,D$ from the symbolic generator developed do not span every single number expressible as a sum of two positive cubes in at least two distinct ways, our method provides a closed-form, algebraic parametrization in terms of a single variable, expressing each term as a radical-exponential function of an integer parameter $c_1$. The generator leverages nested radicals and exponents of algebraic numbers, $α, β$ derived from the recurrence structure of the Diophantine constraint. This work represents the first symbolic, recursive generator of its kind and offers a pathway toward approaching higher powers of this problem from a different lens. These methods exploit structural links between binomial expansions and Diophantine constraints, offering a foundation for extensions to higher powers. |
| title | A Closed-Form Symbolic Generator: $A^n + B^n = C^n + D^n$, for $n = 2,3$ |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2506.19173 |