A Note on the Number of Solutions of the Generalized Ramanujan–Nagell EquationD1x2+D2=4pn

✍ Scribed by Maohua Le

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 668 KB
• Volume: 62
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2019

✦ Synopsis

Let D 1 , D 2 be positive integers with 2 |3 D 1 D 2 and gcd(D 1 , D 2 )=1. Let p be a prime with p |3 D 1 D 2 . In this note we prove that the generalized Ramanujan Nagell equation D 1 x 2 +D 2 =4p n has at most two positive integer solutions (x, n) except (D 1 , D 2 , p)=(1, 7, 2), (3, 5, 2), (1, 11, 3), and (1, 19, 5). 1997 Academic Press 3. (Skinner [8]). If p>2 and D 2 =4p r &1 for some r # N, then N(1, D 2 , p)=2 except N(1, 11, 3)=N(1, 19, 5)=3. 4. (Le [3]). If p>2 and D 2 {4p r &1, then N(1, D 2 , p) 2. 5. (Le [4]). If D 1 >1, then N(D 1 , D 2 , 2) 2 except N(3, 5, 2)=3.