The Diophantine equation 𝑏²𝑋⁴-𝑑𝑌²=1

Bennett, Michael; Walsh, Gary

doi:10.1090/S0002-9939-99-05041-8

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ISSN 1088-6826(online) ISSN 0002-9939(print)

The Diophantine equation

Authors: Michael A. Bennett and Gary Walsh
Journal: Proc. Amer. Math. Soc. 127 (1999), 3481-3491
MSC (1991): Primary 11D25, 11J86
Posted: May 6, 1999
MathSciNet review: 1625772
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Abstract: If and are given positive integers with , then we show that the equation of the title possesses at most one solution in positive integers . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

1. A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439–444. MR 0234912 (38 #3226)
2. M.A. Bennett. On consecutive integers of the form and . submitted for publication.
3. Richard T. Bumby, The Diophantine equation 3𝑥⁴-2𝑦²=1, Math. Scand. 21 (1967), 144–148. MR 0245512 (39 #6818)
4. Zhen Fu Cao, A study of some Diophantine equations, J. Harbin Inst. Tech. 3 (1988), 1–7 (Chinese, with English summary). MR 969961 (90k:11026)
5. Jian Hua Chen and Paul Voutier, Complete solution of the Diophantine equation 𝑋²+1=𝑑𝑌⁴ and a related family of quartic Thue equations, J. Number Theory 62 (1997), no. 1, 71–99. MR 1430002 (97m:11039), http://dx.doi.org/10.1006/jnth.1997.2018
6. J. H. E. Cohn, The Diophantine equation 𝑥⁴-𝐷𝑦²=1. II, Acta Arith. 78 (1997), no. 4, 401–403. MR 1438594 (98e:11033)
7. S. David. Minorations de formes linéaires de logarithmes elliptiques. Publ. Math. Univ. Pierre et Marie Curie 106, Problèmes diophantiens 1991-1992, exposé no. 3.
8. J. Gebel, A. Pethő, and H. G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), no. 2, 171–192. MR 1305199 (95i:11020)
9. J. Gebel, A. Peth\H{o} and H.G. Zimmer. On Mordell's equation. Compositio Math. 110 (1998), 335-367. CMP 98:07
10. Michel Laurent, Maurice Mignotte, and Yuri Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation, J. Number Theory 55 (1995), no. 2, 285–321 (French, with English summary). MR 1366574 (96h:11073), http://dx.doi.org/10.1006/jnth.1995.1141
11. Maohua Le, On the Diophantine equation 𝐷₁𝑥⁴-𝐷₂𝑦²=1, Acta Arith. 76 (1996), no. 1, 1–9. MR 1390566 (97b:11040)
12. D.H. Lehmer. An extended theory of Lucas functions. Ann. Math. 31 (1930), 419-448.
13. David Fog, A theorem on four circles, Mat. Tidsskr. B. 1946 (1946), 113–119 (Danish). MR 0015799 (7,471f)
P. Erdös, On sets of distances of 𝑛 points, Amer. Math. Monthly 53 (1946), 248–250. MR 0015796 (7,471c)
Deane Montgomery and Hans Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. MR 0015794 (7,471a)
Manuel Labra, Calculation of the sides of regular inscribed polygons, Revista Soc. Cubana Ci. Fís. Mat. 2 (1945), 47–67 (Spanish). MR 0015798 (7,471e)
Julio Rey Pasto, The last geometric theorems of Poincaré and their applications, Union Mat. Argentina. Memorias y Monografias (2) 1 (1945), no. 4, 42 (Spanish, with French summary). MR 0015795 (7,471b)
J. Klíma, On some motions of a variable figure in the plane, Věstník Královské České Společnosti Nauk. Třída Matemat.-Přírodověd. 1944 (1944), 5 (Czech). MR 0015800 (7,471g)
H. B. Bone, On orthogonal conic sections, Mathematica, Zutphen. B. 11 (1943), 132–150 (Dutch). MR 0015801 (7,471h)
H. J. Baron, Die Ankugeln des Tetraeders in Beziehung zur Umkugel, Tôhoku Math. J. 48 (1941), 185–192 (German). MR 0015803 (7,471j)
P. Soloviov, Sur un problème géométrographique lié au tracé de la parabole, Nauk.-Doslid. Inst. Mat. Meh. Harkiv. Univ. Geometričniĭ\ Zbirnik 2 (1940), 145–151 (Ukrainian, with French summary). MR 0015802 (7,471i)
Giovanni Gentile, Punti diagonali e poligoni di divisione di un 𝑛-gono piano convesso, Boll. Mat. (4) 1 (1940), 71–74 (Italian). MR 0015797 (7,471d)
14. Wilhelm Ljunggren, Sätze über unbestimmte Gleichungen, Skr. Norske Vid. Akad. Oslo. I. 1942 (1942), no. 9, 53 (German). MR 0011476 (6,169e)
15. Wilhelm Ljunggren, Zur Theorie der Gleichung 𝑥²+1=𝐷𝑦⁴, Avh. Norske Vid. Akad. Oslo. I. 1942 (1942), no. 5, 27 (German). MR 0016375 (8,6f)
16. W. Ljunggren, On the diophantine equation 𝐴𝑥⁴-𝐵𝑦²=𝐶(𝐶=1,4), Math. Scand. 21 (1967), 149–158 (1969). MR 0245514 (39 #6820)
17. Trygve Nagell, On a special class of Diophantine equations of the second degree, Ark. Mat. 3 (1954), 51–65. MR 0061616 (15,854e)
18. R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), no. 2, 177–196. MR 1291875 (95m:11056)
19. N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations, Acta Arith. 75 (1996), no. 2, 165–190. MR 1379397 (96m:11019)
20. Paul M. Voutier, An upper bound for the size of integral solutions to 𝑌^{𝑚}=𝑓(𝑋), J. Number Theory 53 (1995), no. 2, 247–271. MR 1348763 (96f:11049), http://dx.doi.org/10.1006/jnth.1995.1090
21. B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936 (90m:11205)
22. Don Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), no. 177, 425–436. MR 866125 (87k:11062), http://dx.doi.org/10.1090/S0025-5718-1987-0866125-3
23. Wei San Zhu, Necessary and sufficient conditions for the solvability of the Diophantine equation 𝑥⁴-𝐷𝑦²=1, Acta Math. Sinica 28 (1985), no. 5, 681–683 (Chinese). MR 842749 (87e:11042)

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