On the Diophantine equations x~2 - 47y~2 = 1 and ~y2 - Pz~2 = 49

摘要

The integer solution of diophantine equations x2 - Dvy2 = m, (D1 ∈ Z~+,m ∈ Z) and y~2 - D2z~2 = n, (D2 ∈ Z~+, n ∈ Z) is a matter of great concern. Researchers study for different m,n and D1, D2, and obtained some correlation results as follows. When m = l,n = 1, the diophantine equations turns into x~2 - D1y~2 = land y~2 - B2z~2 = 1.At present, there are only a few conclusions on it, see Ref [1] and [2]. When m - 1,n = 4, the integer solution see Ref [3] - [9].Whenm = l,n = 16, the previous conclusions see Ref [10]. When m = 1,n = 49, the diophantine equations turns into x~2 - D1y~2 = land y~2 - D2z~2 = 49. In this case, D1 = 47, D2 can be expressed as 2~tp1~(α1)p2~(α2)p3~(α3)pi~(α4) where a_i = 0 or 1 for 1 ≤ i ≤ 4, and t ∈ Z~+, ps_(1 ≤ s ≤ 4) are different odd primes. Up to now, there is no relevant result on the integer solution of x~2 - 47y~2 = 1 and y~2 - Pz~2 = 49 , this paper mainly discusses the integer solution of it.