Infinite nested radicals
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Barry B. Mittag
Abstract
We prove that each positive integer n can be represented as an integer infinite nested radical and provide an easy method to determine the convergence. The method is straightforward and involves looking at integral quadratic forms. After we proved this, we discovered that the result appears in a somewhat different form in Ramanujan’s Notebooks [1, 2]. However, our proof is so straightforward and the result so striking we think it should be published.
Abstract
We prove that each positive integer n can be represented as an integer infinite nested radical and provide an easy method to determine the convergence. The method is straightforward and involves looking at integral quadratic forms. After we proved this, we discovered that the result appears in a somewhat different form in Ramanujan’s Notebooks [1, 2]. However, our proof is so straightforward and the result so striking we think it should be published.
Chapters in this book
- Frontmatter I
- Introduction V
- Conference booklet and talks VII
- Remembrances of Gilbert XI
- Contents XXIX
- Explicit universal axioms for Kaplansky groups 1
- Isoperimetric and isodiametric functions of group extensions 7
- The homology of groups, profinite completions, and echoes of Gilbert Baumslag 11
- Some properties of the Baumslag groups G(m, n) 29
- Some model theory of the Heisenberg group: I. Unitriangular representations of models of a subtheory of its universal theory 33
- Some model theory of the Heisenberg group: II. The three generator case 47
- On vector-valued Hecke forms 53
- Two algorithms in group theory 73
- On products of closed subsets in free groups 81
- Misbehaved direct products 85
- A survey on Albert algebras and groups of type F4 89
- Perspectives on p-ary bent functions 103
- Multilinear cryptography using nilpotent groups 127
- Musings on generic-case complexity 135
- Noncommutative Gebauer–Möller criteria 149
- On the numbers of the form x2 + 11y2 177
- Separability properties of nilpotent ℚ[x]-powered groups 203
- Infinite nested radicals 219
- Commutative transitivity property in groups and Lie algebras 225
- Simplicial subdivisions and the chromatic number of a group 233
Chapters in this book
- Frontmatter I
- Introduction V
- Conference booklet and talks VII
- Remembrances of Gilbert XI
- Contents XXIX
- Explicit universal axioms for Kaplansky groups 1
- Isoperimetric and isodiametric functions of group extensions 7
- The homology of groups, profinite completions, and echoes of Gilbert Baumslag 11
- Some properties of the Baumslag groups G(m, n) 29
- Some model theory of the Heisenberg group: I. Unitriangular representations of models of a subtheory of its universal theory 33
- Some model theory of the Heisenberg group: II. The three generator case 47
- On vector-valued Hecke forms 53
- Two algorithms in group theory 73
- On products of closed subsets in free groups 81
- Misbehaved direct products 85
- A survey on Albert algebras and groups of type F4 89
- Perspectives on p-ary bent functions 103
- Multilinear cryptography using nilpotent groups 127
- Musings on generic-case complexity 135
- Noncommutative Gebauer–Möller criteria 149
- On the numbers of the form x2 + 11y2 177
- Separability properties of nilpotent ℚ[x]-powered groups 203
- Infinite nested radicals 219
- Commutative transitivity property in groups and Lie algebras 225
- Simplicial subdivisions and the chromatic number of a group 233
