On the numbers of the form x2 + 11y2
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Martin Kreuzer
Abstract
Numbers of the form x2 + Ny2 have been studied for a long time, in particular when N is one of Euler’s convenient numbers. For the first inconvenient number N = 11, most attention has gone to prime numbers of the form x2 + 11y2. Here, we use the class group G11 of level 11 to study the set S1 of all numbers n having a primitive representation n = x2 + 11y2. The four conjugacy classes of elliptic elements of order 2 in G11 give rise to a set C = S1 ∪ S2 in which we have to distinguish S1 from the set S2 of numbers n having a primitive representation n = 4x2 + 11xy + 33y2. The set S1 is decomposed into disjoint subsets corresponding to specific products of primes and cubic numbers. The primes and cubic numbers in S1 and S2 are described in detail.
Abstract
Numbers of the form x2 + Ny2 have been studied for a long time, in particular when N is one of Euler’s convenient numbers. For the first inconvenient number N = 11, most attention has gone to prime numbers of the form x2 + 11y2. Here, we use the class group G11 of level 11 to study the set S1 of all numbers n having a primitive representation n = x2 + 11y2. The four conjugacy classes of elliptic elements of order 2 in G11 give rise to a set C = S1 ∪ S2 in which we have to distinguish S1 from the set S2 of numbers n having a primitive representation n = 4x2 + 11xy + 33y2. The set S1 is decomposed into disjoint subsets corresponding to specific products of primes and cubic numbers. The primes and cubic numbers in S1 and S2 are described in detail.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
