In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.
Citation: Cencen Dou, Jiagui Luo. Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $[J]. AIMS Mathematics, 2023, 8(8): 19353-19373. doi: 10.3934/math.2023987
In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.
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Cencen Dou, Jiagui Luo. Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $[J]. AIMS Mathematics, 2023, 8(8): 19353-19373. doi: 10.3934/math.2023987
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