Pell ve Pell-Lucas sayılarının karşılıklı çarpımsal eşitlikleri ile ilgili Diophantine denklemleri

Yıl 2025, Cilt: 27 Sayı: 2, 464 - 474

Ahmet Emin , Ahmet Daşdemir

https://doi.org/10.25092/baunfbed.1554641

Öz

Bu çalışma, logaritmalardaki lineer formlara dayalı olarak geliştirilen Matveev’s teoremi ve Dujella-Pethő indirgeme lemması kullanılarak, iki rastgele Pell-Lucas sayısının çarpımı olan tüm Pell sayıları ve iki rastgele Pell sayısının çarpımı olan tüm Pell-Lucas sayılarını araştırmayı amaçlamaktadır. Ayrıca, Pell ve Pell-Lucas sayılarına ait ortak terimler incelenmiş ve hiçbir Pell sayısının bir Pell-Lucas sayısının karesi olamayacağı gibi, hiçbir Pell-Lucas sayısının da bir Pell sayısının karesi olarak yazılamayacağı gösterilmiştir.

Anahtar Kelimeler

Pell sayısı, Pell-Lucas sayısı, Diophantine denklemler, Matveev teoremi, Dujella-Pethö indirgeme lemması, logaritmaların lineer formları

Etik Beyan

Yukarıda bilgileri yer almakta olan çalışmamızın tamamı teorik olduğundan yani röportaj, Anket, mülakat, insan ve hayvanlar üzerinde herhangi bir deney gerektirmediğinden ve buna benzer bir sebeple etik kurul izni gerektirmeyen çalışmalar arasında yer aldığını beyan ederiz.

Diophantine equations on cross-multiplicative forms of the Pell and Pell-Lucas numbers

Öz

This study aims to explore all Pell numbers that are the product of two random Pell-Lucas numbers and all Pell-Lucas numbers that are the product of two random Pell numbers based on linear forms in logarithms of algebraic numbers using Matveev's theorem and Dujella - Pethő reduction lemma. Further, we find all the common terms of Pell and Pell-Lucas numbers and show that no Pell and no Pell-Lucas numbers can be written as a square of another.

Anahtar Kelimeler

Pell number, Pell-Lucas number, Diophantine equation, Matveev's theorem, Dujella-Pethö reduction lemma, linear forms in logarithms

Etik Beyan

We hereby declare that our study, the details of which are provided above, is entirely theoretical and does not involve interviews, surveys, experiments on humans or animals, or any similar processes that would require ethics committee approval.

Kaynakça

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