The Brocard–Ramanujan Equation in the Dodecahedral Cubic Field of Discriminant −23
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Pubblicazione: |
Zenodo
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901733306793984 |
|---|---|
| author | ACOSTA PADILLA, ALFREDO LUIS |
| author_facet | ACOSTA PADILLA, ALFREDO LUIS |
| contents | <p>We revisit the Brocard–Ramanujan Diophantine equation <span><span>n!+1=m2</span><span><span><span>n</span><span>!</span><span>+</span></span><span><span>1</span><span>=</span></span><span><span><span>m</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span> from the viewpoint of a specific cubic number field of discriminant <span><span>−23</span><span><span><span>−</span><span>23</span></span></span></span>.<br>In this framework, the factorial is not a scalar but an orbit in <span><span>Z3</span><span><span><span><span>Z</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span></span></span></span> generated by explicit <span><span>3×3</span><span><span><span>3</span><span>×</span></span><span><span>3</span></span></span></span> matrices with determinants <span><span>k3−k+1</span><span><span><span><span>k</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span><span>−</span></span><span><span>k</span><span>+</span></span><span><span>1</span></span></span></span>, arising from the norm form of the field.<span></span><br>The associated Koecher reduction theory produces a dodecahedral polyhedron with symmetry group <span><span>A5</span><span><span><span><span>A</span><span><span><span><span><span><span>5</span></span></span></span></span></span></span></span></span></span>, on which the factorial orbit projects to a finite set of vertices and inscribed tetrahedra.<br>We show that the three known solutions <span><span>(n,m)=(4,5),(5,11),(7,71)</span><span><span><span>(</span><span>n</span><span>,</span><span>m</span><span>)</span><span>=</span></span><span><span>(</span><span>4</span><span>,</span><span>5</span><span>)</span><span>,</span><span>(</span><span>5</span><span>,</span><span>11</span><span>)</span><span>,</span><span>(</span><span>7</span><span>,</span><span>71</span><span>)</span></span></span></span> are precisely the vertices where the cubic norm, the dodecahedral <span><span>A5</span><span><span><span><span>A</span><span><span><span><span><span><span>5</span></span></span></span></span></span></span></span></span></span>–symmetry, and a distinguished elliptic curve <span><span>y2=x3−x+1</span><span><span><span><span>y</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span><span>=</span></span><span><span><span>x</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span><span>−</span></span><span><span>x</span><span>+</span></span><span><span>1</span></span></span></span> intersect, and explain why the orbit cannot hit any further vertex compatible with <span><span>n!+1=m2</span><span><span><span>n</span><span>!</span><span>+</span></span><span><span>1</span><span>=</span></span><span><span><span>m</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span>.<br>This yields a geometric saturation picture: the finiteness of Brocard–Ramanujan solutions is interpreted as a packing limit of factorial torsion in the dodecahedral cubic field.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18808600 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Brocard–Ramanujan Equation in the Dodecahedral Cubic Field of Discriminant −23 ACOSTA PADILLA, ALFREDO LUIS Brocard–Ramanujan equation factorial Diophantine equations cubic number field of discriminant −23 norm forms in number fields Koecher fan and reduction theory dodecahedron and icosahedral symmetry alternating group A 5 A 5 elliptic curve y 2 = x 3 − x + 1 y 2 =x 3 −x+1 orbits in Z 3 Z 3 11D61 – Diophantine equations: exponential and factorial equations 11R04 – Algebraic number theory: algebraic numbers; fields of algebraic numbers 11E76 – Forms of degree ≥ 3; higher degree forms 11F03 – Modular and automorphic forms; elliptic modular forms (context for y 2 = x 3 − x + 1 y 2 =x 3 −x+1) 52B15 – Symmetry properties of polytopes (including Platonic solids and A 5 A 5 –symmetry) <p>We revisit the Brocard–Ramanujan Diophantine equation <span><span>n!+1=m2</span><span><span><span>n</span><span>!</span><span>+</span></span><span><span>1</span><span>=</span></span><span><span><span>m</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span> from the viewpoint of a specific cubic number field of discriminant <span><span>−23</span><span><span><span>−</span><span>23</span></span></span></span>.<br>In this framework, the factorial is not a scalar but an orbit in <span><span>Z3</span><span><span><span><span>Z</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span></span></span></span> generated by explicit <span><span>3×3</span><span><span><span>3</span><span>×</span></span><span><span>3</span></span></span></span> matrices with determinants <span><span>k3−k+1</span><span><span><span><span>k</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span><span>−</span></span><span><span>k</span><span>+</span></span><span><span>1</span></span></span></span>, arising from the norm form of the field.<span></span><br>The associated Koecher reduction theory produces a dodecahedral polyhedron with symmetry group <span><span>A5</span><span><span><span><span>A</span><span><span><span><span><span><span>5</span></span></span></span></span></span></span></span></span></span>, on which the factorial orbit projects to a finite set of vertices and inscribed tetrahedra.<br>We show that the three known solutions <span><span>(n,m)=(4,5),(5,11),(7,71)</span><span><span><span>(</span><span>n</span><span>,</span><span>m</span><span>)</span><span>=</span></span><span><span>(</span><span>4</span><span>,</span><span>5</span><span>)</span><span>,</span><span>(</span><span>5</span><span>,</span><span>11</span><span>)</span><span>,</span><span>(</span><span>7</span><span>,</span><span>71</span><span>)</span></span></span></span> are precisely the vertices where the cubic norm, the dodecahedral <span><span>A5</span><span><span><span><span>A</span><span><span><span><span><span><span>5</span></span></span></span></span></span></span></span></span></span>–symmetry, and a distinguished elliptic curve <span><span>y2=x3−x+1</span><span><span><span><span>y</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span><span>=</span></span><span><span><span>x</span><span><span><span><span><span><span>3</span></span></span></span></span></span></span><span>−</span></span><span><span>x</span><span>+</span></span><span><span>1</span></span></span></span> intersect, and explain why the orbit cannot hit any further vertex compatible with <span><span>n!+1=m2</span><span><span><span>n</span><span>!</span><span>+</span></span><span><span>1</span><span>=</span></span><span><span><span>m</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span>.<br>This yields a geometric saturation picture: the finiteness of Brocard–Ramanujan solutions is interpreted as a packing limit of factorial torsion in the dodecahedral cubic field.</p> |
| title | The Brocard–Ramanujan Equation in the Dodecahedral Cubic Field of Discriminant −23 |
| topic | Brocard–Ramanujan equation factorial Diophantine equations cubic number field of discriminant −23 norm forms in number fields Koecher fan and reduction theory dodecahedron and icosahedral symmetry alternating group A 5 A 5 elliptic curve y 2 = x 3 − x + 1 y 2 =x 3 −x+1 orbits in Z 3 Z 3 11D61 – Diophantine equations: exponential and factorial equations 11R04 – Algebraic number theory: algebraic numbers; fields of algebraic numbers 11E76 – Forms of degree ≥ 3; higher degree forms 11F03 – Modular and automorphic forms; elliptic modular forms (context for y 2 = x 3 − x + 1 y 2 =x 3 −x+1) 52B15 – Symmetry properties of polytopes (including Platonic solids and A 5 A 5 –symmetry) |
| url | https://doi.org/10.5281/zenodo.18808600 |