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On matrices associated to prime factorization of odd integers
In this paper we introduce in section 5 integral matrices M(n) for any factorization of an odd integer n into r distinct odd primes. The matrices appear in several versions according to a parameter ρ ϵ 2 [0, 1]; they have size 2r * 2r and their rank satisfies e.g. for ρ = 1/2 the inequalities of t...
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Інститут фізики конденсованих систем НАН України
2008
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| Series: | Condensed Matter Physics |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/119636 |
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irk-123456789-1196362017-06-08T03:03:08Z On matrices associated to prime factorization of odd integers Bier, T. In this paper we introduce in section 5 integral matrices M(n) for any factorization of an odd integer n into r distinct odd primes. The matrices appear in several versions according to a parameter ρ ϵ 2 [0, 1]; they have size 2r * 2r and their rank satisfies e.g. for ρ = 1/2 the inequalities of theorem 4: r + 1 ≤ rank(M(n)) ≤ 2r⁻¹+1; which are obtained using theorem 1 discussed separately in the first few sections. The cases ρ = 0, 1, 1/2 are analyzed in some detail, and various counterexamples for ρ != 0, 1, 1/2 are included. There are several main results, theorem 5 is a duality between the cases ρ = 0 and ρ = 1, and theorem 6 is a periodicity theorem. The most important result perhaps is theorem 8 (valid for ρ = 1/2 only) on the existence of odd squarefree integers n with r odd prime factors such that rank(M(n)) = r + 1 attains the lower bound shown previously. В цiй роботi у параграфi 5 ми вводимо цiлочисельнi матрицi M(n) для довiльної факторизацiї непарного цiлого числа n на r рiзних непарних простих чисел. Матрицi мають декiлька версiй iндексованих параметром ρ ϵ 2 [0, 1], розмiром 2n * 2n, їх ранг задовiльняє, наприклад, для ρ = 1/2, нерiвнiсть з Теореми 4: r+1... , що одержується за допомогою Теореми 1, яка обговорюється окремо у перших параграфах. Випадки ρ = 0, 1, 1/2 аналiзуються бiльш детально, наводяться рiзноманiтнi приклади для ρ != 0, 1, 1/2. Подаємо ряд головних результатiв: Теорема 5, що описує дуальнiсть випадкiв ρ = 0 i ρ = 1, Теорема 6, що описує перiодичнiсть. Можливо найголовнiшою є Теорема 8 (дiйсна тiльки для ρ = 1/2) про iснування непарних, без квадратiв, цiлих чисел n з r непарними простими множниками, таких, що rank(M(n)) = r + 1, тобто досягає нижньої межi, згаданої вище. 2008 Article On matrices associated to prime factorization of odd integers / T. Bier // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 723-747. — Бібліогр.: 3 назв. — англ. 1607-324X PACS: 02.10.Yn DOI:10.5488/CMP.11.4.723 http://dspace.nbuv.gov.ua/handle/123456789/119636 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we introduce in section 5 integral matrices M(n) for any factorization of an odd integer n into r
distinct odd primes. The matrices appear in several versions according to a parameter ρ ϵ 2 [0, 1]; they have size
2r * 2r and their rank satisfies e.g. for ρ = 1/2 the inequalities of theorem 4: r + 1 ≤ rank(M(n)) ≤ 2r⁻¹+1;
which are obtained using theorem 1 discussed separately in the first few sections. The cases ρ = 0, 1, 1/2 are
analyzed in some detail, and various counterexamples for ρ != 0, 1, 1/2 are included. There are several main
results, theorem 5 is a duality between the cases ρ = 0 and ρ = 1, and theorem 6 is a periodicity theorem.
The most important result perhaps is theorem 8 (valid for ρ = 1/2 only) on the existence of odd squarefree
integers n with r odd prime factors such that rank(M(n)) = r + 1 attains the lower bound shown previously. |
| format |
Article |
| author |
Bier, T. |
| spellingShingle |
Bier, T. On matrices associated to prime factorization of odd integers Condensed Matter Physics |
| author_facet |
Bier, T. |
| author_sort |
Bier, T. |
| title |
On matrices associated to prime factorization of odd integers |
| title_short |
On matrices associated to prime factorization of odd integers |
| title_full |
On matrices associated to prime factorization of odd integers |
| title_fullStr |
On matrices associated to prime factorization of odd integers |
| title_full_unstemmed |
On matrices associated to prime factorization of odd integers |
| title_sort |
on matrices associated to prime factorization of odd integers |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119636 |
| citation_txt |
On matrices associated to prime factorization of odd integers / T. Bier // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 723-747. — Бібліогр.: 3 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT biert onmatricesassociatedtoprimefactorizationofoddintegers |
| first_indexed |
2023-10-18T20:34:57Z |
| last_indexed |
2023-10-18T20:34:57Z |
| _version_ |
1796150581461516288 |