Explicit formula for the solution of simultaneous Pell equations 𝑥²-(𝑎²-1)𝑦²=1, 𝑦²-𝑏𝑧²=1

Cipu, Mihai

doi:10.1090/proc/13802

Explicit formula for the solution of simultaneous Pell equations $x^2-(a^2-1)y^2=1$, $y^2-bz^2=1$
HTML articles powered by AMS MathViewer

by Mihai Cipu PDF

Proc. Amer. Math. Soc. 146 (2018), 983-992 Request permission

Abstract:

For $b$ an odd integer whose square-free part has at most two prime divisors, it is shown that the equations in the title have a common solution in positive integers precisely when $b$ divides $4a^2-1$ and the quotient is a perfect square. The proof provides an explicit formula for the common solution, known to be unique. Similar results are obtained assuming the square-free part of $b$ is even or has three prime divisors.

References

Xiaochuan Ai, Jianhua Chen, Silan Zhang, and Hao 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. MR 3276317, DOI 10.1016/j.jnt.2014.07.009
S. Akhtari, A. Togbé, and P. G. Walsh, Addendum on the equation $aX^4-bY^2=2$ [MR2388048], Acta Arith. 137 (2009), no. 3, 199–202. MR 2496459, DOI 10.4064/aa137-3-1
Michael A. Bennett and Ákos Pintér, Intersections of recurrence sequences, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2347–2353. MR 3326017, DOI 10.1090/S0002-9939-2015-12499-9
Michael A. Bennett and Gary Walsh, The Diophantine equation $b^2X^4-dY^2=1$, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3481–3491. MR 1625772, DOI 10.1090/S0002-9939-99-05041-8
Michael A. Bennett, Mihai Cipu, Maurice Mignotte, and Ryotaro Okazaki, On the number of solutions of simultaneous Pell equations. II, Acta Arith. 122 (2006), no. 4, 407–417. MR 2234424, DOI 10.4064/aa122-4-4
Yann Bugeaud, Claude Levesque, and Michel Waldschmidt, Équations de Fermat–Pell–Mahler simultanées, Publ. Math. Debrecen 79 (2011), no. 3-4, 357–366 (French, with English and French summaries). MR 2907971, DOI 10.5486/PMD.2011.5192
J. H. E. Cohn, The Diophantine equation $x^4-Dy^2=1$. II, Acta Arith. 78 (1997), no. 4, 401–403. MR 1438594, DOI 10.4064/aa-78-4-401-403
Mihai Cipu, Pairs of Pell equations having at most one common solution in positive integers, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 15 (2007), no. 1, 55–66. MR 2396744
Mihai Cipu and Maurice Mignotte, On the number of solutions to systems of Pell equations, J. Number Theory 125 (2007), no. 2, 356–392. MR 2332594, DOI 10.1016/j.jnt.2006.09.016
Nurettin 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), no. 1, 130–136. MR 3529822, DOI 10.1007/s10998-016-0137-0
Pingzhi Yuan and Yuan Li, Squares in Lehmer sequences and the Diophantine equation $Ax^4-By^2=2$, Acta Arith. 139 (2009), no. 3, 275–302. MR 2545931, DOI 10.4064/aa139-3-6
Wilhelm Ljunggren, Zur Theorie der Gleichung $x^2+1=Dy^4$, Avh. Norske Vid.-Akad. Oslo I 1942 (1942), no. 5, 27 (German). MR 16375
Wilhelm Ljunggren, Ein Satz über die diophantische Gleichung $Ax^2-By^4=C\ (C=1,2,4)$, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Institution, Lund, 1954, pp. 188–194 (German). MR 0065575
Florian Luca and P. G. Walsh, Squares in Lehmer sequences and some Diophantine applications, Acta Arith. 100 (2001), no. 1, 47–62. MR 1864625, DOI 10.4064/aa100-1-4
D. W. Masser and J. H. Rickert, Simultaneous Pell equations, J. Number Theory 61 (1996), no. 1, 52–66. MR 1418319, DOI 10.1006/jnth.1996.0137
Maurice Mignotte and Attila Pethő, Sur les carrés dans certaines suites de Lucas, J. Théor. Nombres Bordeaux 5 (1993), no. 2, 333–341 (French, with English and French summaries). MR 1265909
Axel Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284–305 (German). MR 1580770, DOI 10.1515/crll.1909.135.284
A. Togbe, P. M. Voutier, and P. G. Walsh, Solving a family of Thue equations with an application to the equation $x^2-Dy^4=1$, Acta Arith. 120 (2005), no. 1, 39–58. MR 2189717, DOI 10.4064/aa120-1-3
P. M. Voutier, A further note on “On the equation $aX^4-bY^2=2$” [MR2388048; MR2496459], Acta Arith. 137 (2009), no. 3, 203–206. MR 2496460, DOI 10.4064/aa137-3-2
Pingzhi 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), no. 6, 1561–1566. MR 2051114, DOI 10.1090/S0002-9939-04-07418-0