A NOTE ON THE EXPONENTIAL DIOPHANTINE EQUATION (A(2)n)(x) + (B(2)n)(y) = ((A(2) + B-2)n)(z)

Le M., SOYDAN G.

GLASNIK MATEMATICKI, cilt.55, sa.2, ss.195-201, 2020 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 55 Sayı: 2
  • Basım Tarihi: 2020
  • Dergi Adı: GLASNIK MATEMATICKI
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.195-201
  • Anahtar Kelimeler: Ternary purely exponential Diophantine equation, CONJECTURE
  • Bursa Uludağ Üniversitesi Adresli: Evet

Özet

Let A, B be positive integers such that. inin{A, B} > 1, gcd(A, B) = 1 and 2 vertical bar B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A > B-3/8, then the equation (A(2)n)(x) + (B(2)n)(y) = ((A(2) + B-2)n)(z) has no positive integer solutions (x, y, z) with x > z > y; if B > A(3)/6, then it has no solutions (x, y, z) with y > z > x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B 2 (mod 4) and A > B-3/8, then this equation has only the positive integer solution (x, y, z)= (1,1,1).