Faktorizacija s pomočjo stožnic
delo diplomskega seminarja
Abstract
Diplomska naloga preučuje faktorizacijo lihih celih števil, ki jih lahko na dva različna načina zapišemo v obliki $mx^2 \pm ny^2$. V posebnem primeru, ko je $m = n = 1$, sta se s tem problemom ukvarjala že Pierre de Fermat ter Leonhard Euler, katerih rešitve tudi predstavimo. V nadaljevanju te primere posplošimo ter si ogledamo še splošno rešitev Lucasa in Mathewsa. Ker se izkaže, da se negativni primer $mx^2 - ny^2$ precej razlikuje od pozitivnega primera $mx^2 + ny^2$, si ogledamo Pellovo enačbo $x^2 - mny^2 = 1$, ki nam porodi Pellovo povezane rešitve problema. Te pa za razliko od pozitivnega primera dajo trivialen razcep.
Keywords
matematika;faktorizacija;stožnice;Eulerjev razcep;Lucas-Mathewsova formula;Pellova enačba;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2023 |
| Typology: | 2.11 - Undergraduate Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [N. Mrhar] |
| UDC: | 511 |
| COBISS: |
165455107
|
| Views: | 724 |
| Downloads: | 35 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Using Conic Sections to Factor Integers |
| Secondary abstract: | This thesis explores the factorization of odd integers that can be expressed in two different ways as $mx^2 \pm ny^2$. A special case, when $m = n = 1$, was the subject of study by Pierre de Fermat and Leonhard Euler, whose solutions we also present. We continue with a generalization of the problem and present another solution by Lucas and Mathews. As the negative case $mx^2 - ny^2$ turns out to be quite different from the positive case $mx^2 + ny^2$, we take a look at Pell’s equation $x^2 - mny^2 = 1$. We see that Pell-related solutions of the problem produce trivial factorizations. |
| Secondary keywords: | factorization;conic sections;Euler factorization;Lucas-Mathews formula;Pell equation; |
| Type (COBISS): | Final seminar paper |
| Study programme: | 0 |
| Embargo end date (OpenAIRE): | 1970-01-01 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja |
| Pages: | 30 str. |
| ID: | 19956540 |
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