In this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained 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. When a solution exists, assuming the positive solutions of the Pell equation $ x^2(a^2+1)-y^2 = -1 $ 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: Changsheng Luo, Jiagui Luo. Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $[J]. AIMS Mathematics, 2021, 6(9): 9919-9938. doi: 10.3934/math.2021577
In this paper, we consider the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $ where $ a > 0 $ is an integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtained 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. When a solution exists, assuming the positive solutions of the Pell equation $ x^2(a^2+1)-y^2 = -1 $ 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 $.
| [1] | S. Akhtari, The Diophantine equation $aX^4-bY^2 = 1$, J. Reine Angew. Math., 630 (2009), 33-57. |
| [2] |
X. C. Ai, J. H. Chen, S. L. Zhang, H. Hu, Complete solutions of the simultaneous Pell equations $x^2-24y^2 = 1$ and $y^2-pz^2 = 1$, J. Number Theory, 147 (2015), 103-108. doi: 10.1016/j.jnt.2014.07.009
|
| [3] | M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math., 498 (1998), 173-199. |
| [4] |
M. A. Bennett, G. Walsh, The Diophantine equation $b^2X^4-dY^2 = 1$, Proc. Amer. Math. Soc., 127 (1999), 3481-3491. doi: 10.1090/S0002-9939-99-05041-8
|
| [5] |
M. A. Bennett, A. Togbé, P. G. Walsh, A generalization of a theorem of Bumby on quartic Diophantine equations, Int. J. Number Theory, 2 (2006), 195-206. doi: 10.1142/S1793042106000474
|
| [6] |
R. D. Carmichael, On the numerical factors of the arithmetic forms $\alpha^n\pm\beta^n$, Ann. Math., 15 (1913), 49-70. doi: 10.2307/1967798
|
| [7] | M. Cipu, Pairs of Pell equations having at most one common solution in positive integers, An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat., 15 (2007), 55-66. |
| [8] |
M. Cipu, M. Mignotte, On the number of solutions to systems of Pell equations, J. Number Theory, 125 (2007), 356-392. doi: 10.1016/j.jnt.2006.09.016
|
| [9] | M. Cipu, Explicit formula for the solution of simultaneous Pell equations $x^2-(a^2-1)y^2 = 1$, $y^2-bz^2 = 1$, Proc. Amer. Math. Soc., 146 (2018), 983-992. |
| [10] |
J. H. E. Cohn, The Diophantine equation $x^4-Dy^2 = 1$. II, Acta Arith., 78 (1997), 401-403. doi: 10.4064/aa-78-4-401-403
|
| [11] |
J. H. E. Cohn, The Diophantine equation $x^4+1 = Dy^2$, Math. Comput., 66 (1997), 1347-1351. doi: 10.1090/S0025-5718-97-00851-X
|
| [12] | M. T. Damir, B. Faye, F. Luca, Members of Lucas sequences whose Euler function is a power of 2, Fibonacci Quart., 52 (2014), 3-9. |
| [13] | X. G. Guan, Complete solutions of the simultaneous Pell equations $x^2-(c^2-1)y^2 = y^2-2p_1p_2p_3z^2 = 1$, Acta Math. Sin. (Chin. Ser.), 63 (2020), 63-76. |
| [14] |
N. Irmak, On solutions of the simultaneous Pell equations $x^2-(a^2-1)y^2 = 1$ and $y^2-pz^2 = 1$, Period. Math. Hungar., 73 (2016), 130-136. doi: 10.1007/s10998-016-0137-0
|
| [15] |
R. Keskin, O. Karaath, Z. Şiar, Ü. Öǧüt, On the determination of solutions of simultaneous Pell equations $x^2-(a^2-1)y^2 = y^2-pz^2 = 1$, Period. Math. Hungar., 75 (2017), 336-344. doi: 10.1007/s10998-017-0203-2
|
| [16] |
D. H. Lehmer, An extended theory of Lucas' functions, Ann. Math., 31 (1930), 419-448. doi: 10.2307/1968235
|
| [17] | W. Ljunggren, Ein Satz über die diophantische Gleichung $Ax^2-By^4 = C (C = 1, 2, 4)$, (German) Tolfte Skand. Matematikerkongressen, Lund, (1953), 188-194. Lunds Univ. Matematiska Inst., Lund, 1954. |
| [18] | J. G. Luo, P. Z. Yuan, Square terms and square classes in Lehmer sequences, Acta Math. Sin. (Chin. Ser.), 48 (2005), 707-714. |
| [19] |
J. G. Luo, P. Z. Yuan, On the solutions of a system of two Diophantine equations, Sci. China Math., 57 (2014), 1401-1418. doi: 10.1007/s11425-014-4800-8
|
| [20] | P. Ribenboim, The book of prime number records, 2 Eds., New York: Springer-Verlag, 1989. |
| [21] | Q. Sun, P. Z. Yuan, A note on the Diophantine equation $x^4-Dy^2 = 1$, J. Sichuan Univ., 34 (1997), 265-268. |
| [22] |
A. Togbe, P. M. Voutier, P. G. Walsh, Solving a family of Thue equations with an application to the equation $x^2-Dy^4 = 1$, Acta Arith., 120 (2005), 39-58. doi: 10.4064/aa120-1-3
|
| [23] |
P. M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comput., 64 (1995), 869-888. doi: 10.1090/S0025-5718-1995-1284673-6
|
| [24] | P. Z. Yuan, A note on the divisibility of the generalized Lucas sequences, Fibonacci Quart., 40 (2002), 153-156. |
| [25] |
P. Z. Yuan, On the number of solutions of $x^2-4m(m+1)y^2 = y^2-bz^2 = 1$, Proc. Amer. Math. Soc., 132 (2004), 1561-1566. doi: 10.1090/S0002-9939-04-07418-0
|
Changsheng Luo, Jiagui Luo. Complete solutions of the simultaneous Pell equations $ (a^2+1)y^2-x^2 = y^2-bz^2 = 1 $[J]. AIMS Mathematics, 2021, 6(9): 9919-9938. doi: 10.3934/math.2021577
DownLoad: