On the Diophantine Equation $left(9d^2 + 1right)^x + left(16d^2 - 1right)^y = (5d)^z$ Regarding Terai's Conjecture

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Alan M., T.

Journal of New Theory, sa.47, ss.72-84, 2024 (Hakemli Dergi) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2024
  • Doi Numarası: 10.53570/jnt.1479551
  • Dergi Adı: Journal of New Theory
  • Derginin Tarandığı İndeksler: TR DİZİN (ULAKBİM)
  • Sayfa Sayıları: ss.72-84
  • Yıldız Teknik Üniversitesi Adresli: Evet

Özet

This study proves that the Diophantine equation $left(9d^2+1right)^x+left(16d^2-1right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy's Primitive Divisor Theorem, specifically when $d$ is not divisible by $5$. In cases where $d$ is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.